Kara has been easily soloable by a lot of classes since 85, so I expect it's still true at 90.
Four bosses drop pets: Terestian Illhoof, The Curator, the Big Bad Wolf (part of the Opera Event), and Malchezaar
The Big Bad Wolf is the first boss you'll encounter that drops a pet. The Opera Event is a required Boss, but you have to defeat Moroes before you can do it. Which Opera Event you get is random, so this one will probably cause the most fits for people.
The Opera Event is not set, so you can run multiple alts each week hoping that one or more of them gets the Big Bad Wolf.
The Curator is the next required boss after Opera Event.
Illhoof is an optional boss after the Curator. The "secret" entrance to his spot is obvious by the imps hanging out in front of it.
The next optional boss is Shade of Aran. I recommend killing him because:
Doobjanka wrote:Pro-tip for Karazhan: If you plan to go back later for Chess event, kill Shade of Aran for the easy teleport. Saves five or six minutes of running later.
Malchezaar is the final boss. You have to get through Chess Event to get to him.
Chess Event strat that works for me 50% of the time (the cheating RNG makes very few strats foolproof):
Move the pawn in front of the king out for when Medivh cheats the first time
Move the pawns in front of the "knights" out so that you can move the two rooks diagonally two spaces. Prioritize which rook you move first based on which side Medivh is moving pieces on.
Move the king out of fire when you have to
Use either the queen or king to help finish off the opposing king
I've created another thread that shows the strat in picture form
Malchezaar is more or less a zerg at this point.
Required Boss Path:
Moroes > Opera Event > Curator > Chess Event > Malchezaar
Kara Pro Tip: There are mana wyrms that you can't kill with spells that may get at you. You also can't kill them with wands. Make sure you have a dagger or staff available to use as a weapon if you aggro one as a caster.
There are no required bosses in Serpentshrine Cavern; you can tackle whichever one you want whenever you want (which is good when you only need one more pet from the instance).
The bosses that drop pets are Morogrim Tidewalker, Hydross the Unstable, and Lady Vash'j. I believe that for most classes Tidewalker and Hydross are zergable at 90. Vash'j has a complicated fight mechanic that may take you a few attempts to figure out (see Wowhead comments for potential class soloing recommendations).
The good news is you don't have to kill Kael'thas solo to get a pet (still can be a quite complicated at 90, altho there are plenty of soloing strats available since he drops Ashes of Al'ar).
Like SSC, the bosses do not have to be done in any order. Al'ar, Voidreaver, and Solarian drops the pets. All three bosses have complicated fight mechanics but I don't believe any are so bad as to prevent soloing at 90 (if you're concerned, just check the respective boss pages at wowhead).
I hope this was somewhat helpful. Feel free to post any strats, etc., in this thread.