In Mass Effect 3, What are the Race/Class Combinations & Hidden Bonus ones?

Copied exactly from the Mass Effect 3 Wikia, included with the powers per races:

Adept

• Human: Warp, Singularity, Shockwave, Alliance Training, Fitness (Adept)
• Asari: Stasis, Warp, Throw, Asari Justicar, Fitness (Adept)
• Drell: Reave, Pull, Cluster Grenade, Drell Assassin, Fitness (Adept)

Soldier

• Human: Adrenaline Rush, Concussive Shot, Frag Grenade, Alliance Training, Fitness** (Soldier)
• Krogan: Carnage, Fortification, Inferno Grenade, Krogan Berserker, Rage
• Turian: Marksman, Concussive Shot, Proximity Mine, Turian Veteran, Fitness (Soldier)
• Battlefield 3:1 Adrenaline Rush, Carnage, Frag Grenade, Alliance Training, Fitness (Soldier)

Engineer

• Human: Incinerate, Overload, Combat Drone, Alliance Training, Fitness (Engineer)
• Quarian: Incinerate, Cryo Blast, Sentry Turret, Quarian Defender, Fitness (Engineer)
• Salarian: Incinerate, Energy Drain, Decoy, Salarian Operative, Fitness (Engineer)

Sentinel

• Human: Throw, Warp, Tech Armor, Alliance Training, Fitness (Sentinel)
• Turian: Warp, Overload, Tech Armor, Turian Veteran, Fitness (Sentinel)
• Krogan: Incinerate, Lift Grenade, Tech Armor, Krogan Berserker, Rage

Infiltrator

• Human: Sticky Grenade, Tactical Cloak, Cryo Blast, Alliance Training, Fitness (Infiltrator)
• Salarian: Energy Drain, Tactical Cloak, Proximity Mine, Salarian Operative, Fitness (Infiltrator)
• Quarian: Sticky Grenade, Tactical Cloak, Sabotage, Quarian Defender, Fitness (Infiltrator)

Vanguard

• Human: Biotic Charge, Shockwave, Nova, Alliance Training, Fitness (Vanguard)
• Drell: Biotic Charge, Pull, Cluster Grenade, Drell Assassin, Fitness (Vanguard)
• Asari: Biotic Charge, Stasis, Lift Grenade, Asari Justicar, Fitness (Vanguard)

You can be a Human male or female in any of the 6 classes, and the Human options are unlocked from the start. Alien races are listed below, and are only unlockable via the Veteren item pack or better in the store.

Adept: Asari, Drell
Engineer: Quarian, Salarian
Infiltrator: Quarian, Salarian
Sentinel: Krogan, Turian
Soldier: Krogan, Turian. There's also a bonus Battlefield 3 class if you sign into Origin with an account that has a Battlefield 3 online pass activated.
Vanguard: Asari, Drell

A nice table is available on the Mass Effect Wiki as well.