I don't really find them to be very gamemode specific (only trophy system seems to be most usefull in Dom, Momentum, and similar modes than in TDM or KC for example). The one you use is more a play style preference.
I think they are more useful in hard core since the effects help you stay alive and with less health it's very useful to have some kind of buff.
My favorite is overclock but sometimes I use tactician wildcard (you can use 2 abilities at once doing that ex: cloak and hover makes you fly invisible).
If I was going to suggest an exo ability (and since alot of people compared this game to halo) I would add the hologram abilitiy from halo. Basically you press the button and a hologram replica of yourself runs straight toward whatever you were looking at. On halo it didn't buff any stats but it was always fun to send holograms into a room full of enemies then shoot them in the back while they shoot at the hologram. It was also cool seeing people freak out when an army of players and holograms swarmed them.
I use exo mute when playing snd and snd is all i play haha. if i'm in a high action area i use it or on my way to plant the bomb
I have actually started playing allot more without them, because I forget to use them mid match and would rather use that slot for something else.
but before I use to use the shield and hover allot.
Up until the recent past I used nothing but EXO ping. I would spawn in and ping to check on near by threats, and look for people stationary and laying on the ground, I would find a lot of campers and people controlling streaks that way.
Now I am using EXO Trophy System. I find it helps a lot more. Have you been in a game, and seen the first few minutes nothing but a frag fest? I don't worry about it with the Trophy System. Once people start to see that the boom sticks are not working as expected they change tactics. I can't tell you how many times I have seen someone pop up and tube me, only to see their grenade vanish as I shoot them before they can swap guns. When I am sniping I have seen Stinger groups dissipate, and have been able to get the guy who launched them.
I have been playing HC mostly lately, and can say that there is on ability that I think will only be worthwhile on core. To get the stim achievement you have to kill people with stim while taking damage. There is not a lot of damage in HC to be taken.
Do you find a certain exo ability more useful in differing game modes ? Hardpoint and Momentum I suggest running Exo Tactician as a wildcard and run the Exo Trophy system and Overclock. Search and Destroy I suggest running Exo Mute. Domination I would suggest running Overclock.
Do you find a certain exo ability more useful in Hardcore or Core ? There is not much difference IMO regarding HC and Core when it comes to Exo abilities
Do you have a favorite exo ability just because it's fun ? I think its fun to surprise your enemy by running the Exo Cloak