Which support ships? In what quantity?

It depends quite a bit, but the Hoshilo (TEC), Iconus Guardian (Advent), and Overseer (Vasari) are the main defensive ships you'll want to have around to increase your fleets survivability. The Vasari Subverter is also a great offensive weapon in trained hands for its ability to shut down entire fleets. The Subjugator and Celio are not usually built, and if they are only in limited numbers.

Quantity is a much harder thing, you've really got to get a feel for what goes well, and that to a degree depends on how well you can manage your fleets.

If you want to be pro you need to spam hoshiko, research demobots and put them on auto. Srsly.

Meh, Overseers kind of bite the big one though.  They're the least efficient in terms of cost, supply, and deployment time.  They also need to face the target to provide their heal.  If you want to use them, you need to carefully micro them at all times, or they won't be of much use: despite having an instant 200 hp heal.  If their phase disruption were passive like their mobile detection then it might be worth having some more mixed into battle groups. 

The Subjugator recently got a buff where it no longer has to face a target. 

The Cielo has a minor buff on its own, but can be worth having a few around if you have Designate Target.  -40% damage absorbtion works wonders when Ogrovs, bombers, or LRMs need to blast something into oblivion in a hurry. 

For the last guy to post, it's more like spam Arcovas with Timed Charges automatically set to auto-cast when researched and watch them plant bombs on invulnerable extractors in neutral zones.  Hello wasted 15 minute cooldown time.  Thanks, scripting oversight. 

When it comes to protecting capital ships and starbases from focus fire, though, they're without peer.  Definitely well-worth your time in any situation where you expect to have your capital ship focus fired... which is pretty much always.

Generally speaking, this is one auto-cast ability you want to turn off.  Even when it does get a target right, you'll often get overkill where too many scouts will use the ability at once and waste it.

Meh, I find people will just focus fire the Overseer down once they see it/them.  So at best they dump their antimatter and heals on each, and still explode into space debris. 

And what I was saying is that if your scouts are parked somewhere with a neutral extractor when you research the tech, they'll instantly auto-cast it if you don't catch them fast enough.  That was really dumb, all around.  But yes, I normally flick that off, but it still shouldn't be set to auto-cast by default when it has a 15 minute cooldown time. 

They can shoot your overseers out but other overseers will still heal one that is getting focus fire and in the mean time rest of your fleet is killing his fleet ........

Well, I guess you can at least make them try to decide between that Skirantra and the Overseers.  Though fleets seem to get large enough that any individual ship might explode in seconds.

If you micro well enough overseers can heal 1000s of HP in mater of seconds.

All other races are time dependent regarding healing. This is where overseers really shine.

1000s of HP in seconds is no exaggeration...I've even seen the AI keep a single frigate alive under the full fury of a fleet just by using overseers...truly great ships, and microing them is certainly much easier than microing subverters or advent support ships...

As Mecha Lenin said though, hoshikos with dem bots are the way to go now...the only counter to them is LFs with abilities, which would be a great counter if it weren't so hard to keep LFs alive...a TEC fleet with LRMs, Hoshikos, bombers, and some flak is going to be pretty damn difficult to defeat...Vasari can at least try to use subverters, and they do have much better SC...as for advent, I'm pretty sure now Advent is inferior to TEC fleets late game...repulsion certainly helps but all it does is delay the inevitable....

Alright, you got me.  They are amazing when the late game comes around, usually in team settings.  Combine them with Reintegration on Enforcers and Repair Clouds from Skirantras and your ships can survive a good amount of focus fire. 

I have to keep in mind that you guys tend to play large team games, and that certainly changes things.  I was thinking more of a 1v1 on a smaller scale.  Overseers are much harder to deploy in effective numbers there, unless the other person is being lax and asking for a loss. 

1v1s do change things...

In those cases, early game tactics tend to be more important unless it is a large map or some custom symmetrical map that causes the game to drag on...in that case, Vasari are better off deploying SBs, rushing with skirantras, or some creative little strategy...

Overseers...well, they pretty much suck until you hit a certain threshold, but then they are amazing...in a 1v1 though, I think you are right, it wouldn't be very likely that a player would ever amass that many overseers unless it was a real odd situation...

Let me sum this up as best as I can, using the large fleet vs fleet scenario:

Vasari:

Stilakus Subverter: Around 10 or so of these. Keep them close to the front lines, spread out from each other. Micro them to get close to the packs of enemy ships and let them do their jobs.

Overseer: 2 to 3 at most. Make sure it is the ship you have least of. Keep them close to your most critical ships and when said ships get critically damaged, micro time.

Advent:

Guardian: Get about 6 of these in a spread out formation, and when the enemy opens fire, they will automatically use their abilities.

Domina Subjugator: This ship can be very powerful if in the right hands. It can disable ships for a considerable duration with some sort of ability. Use 2 to 3 at most against the enemy's most critical ships.

TEC:

Hoshiko Robotics Cruiser: Get a decent bunch, roughly 1 for every 4 to 6 other ships in your fleet and let them do what they do. Make sure Demobots are researched!

Ceilo Command Cruiser: Use 2 or 3 of these. Embolden doesn't really do much when a ship is under direct fire, I've noticed, but Designate Target, as stated above and proven by me and the expierenced player, works wonders against Starbases and Caps if you have LRMs, Ogrovs, Bombers, and don't forget the Marza Dreadnought.

Enough said.