Of the six classes, this is what I usually roll with (bear in mind I do not have all characters unlocked yet... for example, I can't speak about the Vorcha because I don't have one

):
Adept
Ex-Cerberus with Carnifex, specced for biotic explosions. Lash and Smash are great fun. This is my second favorite character.
Soldier
Krogan with Mattock and full Carnage. Seriously can't not laugh at the wiggly tail, the headbutts, and the "MRWAHAHAHA!"
Engineer
Either Geth with Carnifex specced for health/shields, no Hunter Mode, healing turret... or Male Quarian with Carnifex and full Tactical Scan and full Arc Grenades.
Sentinel
Turian with Mattock, full health/shields, maxed Warp for damage, maxed Overload for chain/shields, and 4 points in Tech Armor for damage resistance. This is my absolute favorite, not necessarily because it's the most fun (i.e. Ex-Cerberus and Krogan Soldier have more of a fun factor because of the rambo craziness you can pull off), but because it is my most efficient at dealing with any enemy type and it is my most comfortable as far as "this is the roll I was born to fill" kind of way.
Infiltrator
Geth with either Geth Plasma Shotgun or Reegar Carbine. Full Tactical Cloak and Hunter Mode. Can only play a few matches before Hunter Mode gives me a major headache, so not the most fun because of that (although the character itself is fun to play and I like Hunter Mode... just not the intense brightness).
Vanguard
Either Human Female with Carnifex and fully specced Biotic Charge and Nova... or Krogan with Reegar Carbine fully specced with everything except Carnage.