I've played mainly as the Advent, because they best fit my playing style. Here are my observations on their capital ship abilities so far:

Radiance Battleship
Detonate Antimatter: Quite a good ability, can deal a fair amount of damage when leveled up. Useful for depleting    antimatter stores on enemy ships.
Animosity: An ability useful during the early game, when you need the Radiance to draw fire from your weaker frigates. I have found little use for it late game, except used with the Rapture's Vengeance.
Energy Absorption: I have found that this is quite useful in keeping your Radiance alive. At level 3, this converts 15% of all damage done to your Radiance into antimatter, which also lets you use abilities more often. This is great if used with the Revelation's Guidance, which reduces ability cooldowns. It also gives a boost to armor, which the Advent are lacking in.
Cleansing Brilliance: A very good level 6 ability, doing a lot of damage when used at the right time. If used in conjunction with the Progenitor's Malice, this can wipe out entire enemy fleets. However, with the changes made to Malice in patch 1.05, this requires careful timing.

Revelation Battlecruiser
Reverie: A  good ability that disables all weapons and abilities on an enemy ship, with a good 40 second duration at level 3. If enough damage is done to the target though, it will be shaken out of its "trance".
Guidance: A good ability only when you have multiple capital ships in a fleet. This can let them use abilities more often by reducing their cooldown.
Clairvoyance: A good spying tool, with a decent 150 seconds duration at level 3. However, a careful opponent can spot the "eye" above their planet, so be careful when using this to plan attacks.
Provoke Hysteria: Another great level 6 ability that can destroy enemy planets very quickly but the duration is rather short in my opinion. However, this ability's duration carries on even after you destroy an enemy colony and colonize it. I have accidentally destroyed some of my own colonies because if this, so be careful.

Progenitor Mothership
Colonize: Pretty self-explanatory, comes with a reduction to structure build time and planet upgrade costs.
Malice: A super power in my opinion, it spreads damage to all ships within range. Can increase your fleet's damage by over 300% in long battles. This can destroy entire enemy fleets if used with the Radiance's Cleansing Brilliance.
Shield Regeneration: Good for keeping your ships alive.
Resurrection: A good ability for players who find themselves losing high level capital ships; it lets you bring a lower level one back up to that level.

Halcyon Carrier
Telekinetic Push: Good if your opponent has lots of strike craft, this pushes them away from your ships along with doing some damage to them.
Adept Drone Anima: A so-so ability, increasing the number of drones in each squadron.
Amplify Energy Aura: Another super power. This decreases the weapon cooldown on all energy weapons (which are the only ones the Advent use), in other words increasing the DPS of all ships within range.
Anima Tempest: Another so-so abiltiy, sends extra strike craft to fight for a while.

Rapture Battlecruiser
Vertigo: Quite a good ability, increases weapon cooldown on enemy ships and reduces their chance to hit.
Vengeance: A powerful ability that can retaliate 120% damage at level 3. Very powerful if used with the Radiance's Animosity.
Concentration Aura: Good only if you have a lot of strike craft, this increases their damage if within range.
Domination: Not worth the antimatter cost in my opinion, this permanently brings an enemy frigate under your control.

All comments and opinions are welcome. If there is anything I have missed out, let me know and I will edit it. I would be curious to know how these abilities compare with those on TEC and Vasari capital ships.