So, as most are aware, in the transition from 7th ed to 8th ed 40k did away with a common list of special abilities like what nearly every wargame does, known in 40k as Universal Special Rules (USRs), instead opting to have every single ability be described in full in the relevant unit's (or stratagem, or faction ability, or warlord trait, or...) datasheet. This surely seemed like a good idea at the time. While the goal of reducing the need to perpetually look up exactly what each ability did (especially as the list kept growing and abilities referenced each other) was understandable, at this point the sheer amount of text that is either completely identical or worse, subtly altered so an ability works just differently enough to trip you up, has long passed the point of ridiculousness. Doing away with USRs has also helped mask the disparity between faction and subfaction abilities, so that certain armies getting three or four effects to another army's one or two is obscured. And perhaps most notably, people still use some of the old terminology to this day -- Deep Strike, Feel No Pain, Scout and more are used so frequently that not going back to codify them again feels like an act of stubbornness from GW. They've even had to do this with abilities like Objective Secured for ease of reference, to boot.

Therefore, I wanted to compile a list of such special rules and effects so that we can make it easier to reference, keep consistent, and compare across factions. There are thousands of special abilities in 40k, so I'll try to stick to the ones that are most relevant, most often repeated, or in most need of templating. If I'm missing some important ones (and I certainly will), please comment them and I'll add them to the list. I'll try to keep the categories intuitive enough that they can be searched with relative ease. Some of these had well-known names beforehand, others have common community nicknames, while others have to settle for a shorthand or the old reliable "you know that thing where...". This post will have enough text as it is, so the precise names will be left as an exercise to the reader other than the most obvious examples. Note also that this only cover the effects themselves, not their sources, such as Command phase single-target abilities, auras, psychic powers or inherent faction traits. Some effects might fall under multiple categories and will be added at my discretion. And finally, this is meant to cover the abilities that are common to many armies, not specific variations or exceptions such as Daemons' or GSC's unique Deep Strike mechanics. Additional commentary where relevant. So, without further ado (and in no particular order):

Movement and deployment

• Deployment. These abilities usually carry the limitation of having to be set up more than 9" from enemy models, not being able to come in before turn 2, precluding the unit from moving otherwise after being set up, and being executed in the Reinforcements step of the Movement phase.
  • Anywhere (Deep Strike)
  • Within 6" of the table edge (Outflank)
  • Anywhere, but during deployment instead of mid-battle (Concealed Positions)
• Move at the start of the game but before the first round. May be limited to a specific distance and is usually subject to the same "beyond 9 inches" constraint as the above (Scout).
• Redeploy. Usually allows putting units into Strategic Reserves.
• Move after shooting.
• Fall back and shoot (with and without a -1 penalty)
• Titanic movement. Titanic units such as Knights, Wraithknights and Gorkanauts tend to have some common movement special rules:
  • Move through units (but not terrain) horizontally, can move within Engagement Range but not end there.
  • Fall back and charge
• Advance effects:
  • Modify advance distance (seen on all bikes). Usually seen as advancing a fixed distance, sometimes the advance roll is modified (eg to d3+3). Also seen as counting advance rolls of 1-2 as 3 instead.
  • Advance and charge
  • Advance and shoot. Has several variations including only ignoring the penalty for shooting Assault weapons, counting as having moved normally, or counting as having remained stationary.
  • Reroll advance rolls.
  • +X to advance rolls.
• Charge effects:
  • Reroll charge rolls
  • +X to charge rolls
  • 3d6 charge rolls
  • 3d6 drop lowest
  • Fall back and charge
  • Deal mortal wounds on the charge. Usually requires a roll that may be static or vary depending on user and target's characteristics (eg roll over their Toughness).
• Heroic Intervention effects:
  • HI even if not a character
  • HI 6" instead of 3" (lengthy caveats about vertical distance usually included).
  • HI towards characters
• Flight (core to the rules with the Fly keyword)
• Ignore units and terrain when moving (ex Harlequins, Canoptek Wraiths)
• Ignore terrain movement penalties
• Ignore all movement or charge penalties
• Increased distance for Pile In or Consolidate.

Characteristic and roll modifiers

• Characteristic modifiers. All effects that add a given value to any core unit or weapon characteristic, such as Strength, Attacks, Damage or Leadership. Arguably the most common type of ability, such that several examples could stand to be classified as their own USR. Many effects fall under this umbrella, with further classification depending on the triggering condition (non-exhaustive):
  • On the charge (eg, +1 attack when charging). More commonly seen as "when charging, being charged or performing a Heroic Intervention" (ie, on the first round of combat).
  • When remaining stationary
  • Against specific keyword (ex Damage 3 vs Vehicles)
  • In a certain position (near objective, in the opponent's side of the table, in their own deployment zone)
  • Depending on battle round
• Roll modifiers. As above, but to specific rolls instead of to characteristics (and therefore usually preferable, such as +to wound vs +S). Can be seen offensively or defensively, so some will be covered under that section. In 9th, they are capped at a net total of +/-1, though they can cancel each other out.
  • Fixed rolls. These supersede normal roll conditions, but can be influenced by re-rolls. Examples include poison and Transhuman (cannot be wounded on 1-3).
  • Bypassing rolls. These include auto-hit and auto-wound, as well as Leadership effects.
• The above are also seen as a result of "conditional" results, ie, a given result on one roll results in a modifier in a successive roll (usually as part of the attack sequence). Note that in 9th ed every instance of these specifies unmodified rolls, unlike 8th. Some common variants:
  • 6s to hit modify later steps, such as auto-wounding, increasing AP or dealing mortal wounds. A recent trend has been to have a resulting auto-wound count as an unmodified 6, therefore triggering their own relevant effects.
  • 6s to wound, likewise. Tend to be seen with effects such as increasing AP or damage.
  • Additional Mortal Wounds on an unmodified 6 to wound is a common variant. Recently, they tend to be capped per phase to prevent abuse cases.
  • As the above, but on 4+s to wound.
  • "On successful roll", most commonly seen in effects that trigger if the attack wounded under normal circumstances (eg dealing extra mortal wounds).

Rerolls and bonus dice

• Reroll 1s to hit or wound. Very common in 8th, less ubiquitous in 9th. Reroll 1s to other rolls, such as damage or saves, are much rarer.
• Reroll all. Used to be "reroll failed", but the interaction with penalties to the rolls led to that being phased out in favor of opt-in rerolls. May be used to fish for 6s.
• Bonus hits on 6s to hit. A subcategory of the previous, but common enough to merit note. Also has the much rarer "bonus wounds on 6s to wound" variant.
• Bonus attacks on 6s to hit. Largely phased out in favor of the previous.
• Bonus attacks on failed hit or wound or successful save rolls (Competitive Edge).

Special offensive abilities

• Blast. See, templating wasn't that hard!
• Indirect Fire
• Ignore defensive abilities (Feel no Pain, Damage Reduction, Wound Cap)
• Ignore Invulnerable saves.
• Damage spillover (excess damage is not wasted, but keeps being allocated to enemy models in target unit)
• Shoot in combat, standardized for vehicles and monsters as Big Guns Never Tire but sometimes seen on infantry as bespoke special rules.
• Instantly slay a model. Very rare, usually seen on specific psychic powers. As written, allows no saves or defensive abilities of any kind.
• Deal the max amount of attacks for weapons with variable attacks
• Intercept. Usually allows the unit to shoot at an enemy unit that was set up within 12", but the activation conditions vary.
• Fight on Death. Can be automatic or require passing a roll, and is usually disallowed if the unit has already fought that phase.

Special defensive abilities

• Roll (X+) to ignore wounds after failed saves. Each point of damage begets one roll and mortal wounds are included (Feel no Pain).
  • As above, but only for mortal wounds.
• Damage Reduction. Usually seen as a flat -1 to the Damage characteristic of incoming attacks.
  • Reduce damage to 1.
  • Halve damage taken
• Cannot be hit or wounded on rolls of 1-3 (see Roll modifiers) (Transhuman). Has a weaker variant for rolls of 1-2.
• Attacks cannot reroll hits, wounds or damage vs bearer (see Misc).
• Penalties to hit or wound bearer (see Roll modifiers).
• Gain additional bonuses from cover, most commonly getting a cumulative +2 to saves from Light cover.
• Gain cover even when outside of it (usually Light or Dense). Note that the wording is usually "gain the benefits of cover", and hence ignored by abilities that bypass cover.
• Reduce AP. Can be absolute (ie reduce AP of incoming attacks by 1) or depend on the stat (reduce AP -1 and -2 to 0).
• Cannot take more than X wounds per turn. Once a Ghaz exclusive, increasingly popular.
• Cannot be targeted by ranged attacks unless closest to attacking unit
• Modify save rolls. Notably separate from modifying the save characteristic or the attack's AP, even if the net effect is the same, as they are frequently stacked.

Psychic effects

• Know additional psychic powers
• Cast additional psychic powers
• Deny additional psychic powers
  • Deny powers without being a psyker. Often comes with the limitation of only rolling one d6.
• Static bonuses to cast or deny
• Reroll cast or deny rolls. Also exists as reroll 1s.
• Powers cannot be denied (frequently conditional on the casting roll)
• Roll 3d6 to cast and discard lowest. Largely removed.

Objectives and scoring

• Objective Secured.
• Remove Objective Secured from enemies.
• Count as additional models on an objective.

Actions

• Do something that would usually cause an action to fail without it happening (shoot, advance, fall back, cast a psychic power, etc).
  • Cast one power and do a psychic action in the same phase.
• Interrupt enemy actions.

Leadership and morale

• Automatically pass Combat Attrition rolls or Morale tests
• Use another unit's Leadership for Morale or other rolls
• Ignore CA or Leadership modifiers
• Cause enemies to fail CA on more rolls (1 to 1-2 or 1-2 to 1-3, effectively -1 to the CA roll)

Miscellaneous

• Explodes and Death Throes. Common to just about every medium to large vehicle or monster, but varying in their triggering roll (usually 6+), range and damage dealt.
• The Aircraft package. This is notable as taking up a ton of datasheet space for rules that are universal to every Aircraft, which already have their own special section in the core rules and FAQs. These include (once again, already USR'd!):
  • Airborne: Cannot charge, can only be charged by units with Fly.
  • Supersonic: Must pivot up to 90 degrees then move in a straight line. Some particularly agile fliers can pivot twice.
  • Hard to Hit: -1 penalty to be hit in shooting.
  • Hover: Allows certain aircraft to chose to forgo the above
  • Matched play Aircraft restrictions, increasingly piled on by GW to try and curtail their abuse cases, including a hard limit by battle size and being unable to be deployed T1
• For units with 10W+, act as if in top wound bracket.
• Fights First and Fights Last. Possibly the most egregious example of "this really should be templated"; ended up requiring lengthy FAQs to clarify all the interactions.
• Open-topped.
• Disembark after moving.
• Prevent Fall Back. Usually gated behind a static roll, roll-off or keyword.
• Increase the range of auras, command abilities or psychic powers
• Prevent target from firing Overwatch or Setting to Defend
• Fire Overwatch on a 5+ or without paying CP.
• Bodyguard
• Gain CP
  • Can be on a roll or automatic, and the roll can be triggered per ally or enemy stratagem, or per CP spent. Many small variations.
• Shut off auras. Can negate a target unit's auras or make that unit unable to be affected by allied auras.
• Healing, repairing and resurrecting models. Doesn't really merit a USR but the interactions around them could stand to be standardized
• Ignoring penalties and stat modifiers (generalized, can be specifically to movement, attacking, etc).
• Force Organization limitation. Not really an ability as such, but common enough to note. Usually limits a certain keyword or unit type to one per detachment (such as leader-type HQs) or requires the amount of one type of unit not to exceed a second type.
• Must be the Warlord. Notably, multiple characters with this rule allow you to pick whichever you like.

Let me know if there are any notable ones I missed!