Surprisingly enough, I had a conversation about this with a friend of mine not too long ago. My idea came in the form of three SPECTRE classes that accentuated what each of the Council Races would value as an exemplar of their respective combat tactics. Putting them in the big three "pure" classes of Soldier, Adept, and Engineer would really help set them apart as well if you wanted to go that far.
Turian SPECTRE:
Turian battle tactics seem to really favor precision over anything else. Waiting to catch your enemy with his pants down so you can take that one shot to keep him down. So you'd want to start with an "activate" power that really showed that off, so I'd suggest Marksman to keep even your automatic weapons where you're pointing them. Secondly, you'd want an ability that would take care of an enemy's defenses, for this I'd either suggest a trusty Concussive Shot or Overload to rip apart that stood between you and your mark's grave. After that, I'd suggest a type of grenade. Frag or Sticky would be the perfect tool to root out anything behind cover too thick to shoot through, not to mention you could clear a room without having to get your hands dirty.
Asari SPECTRE:
I know this one seems like a no brainer on the surface. "Of course it'd be a Biotic class, ALL Asari are Biotics." But the real issue with this one is making is stand apart from the other Asari classes. First off, I'd suggest deciding on whether it'll be a front like or support character. If it's a front line class, give it Barrier. It's a solid ability that works just like the Sentinel Tech Armor. If you wanted a more back row, support class, I'd be as bold as to suggest Dominate, Morinth's unique power from ME2. We don't see it in ME3, so it'd be unique and really show this Asari as having a personal flair. After that, you'd just decide between your choice of Biotic Explosion. So either a Warp/Shockwave with Barrier to make a very clear front line class, or a Stasis/Throw with Dominate to make a master crowd control class.
Salarian SPECTRE:
Now this guy should be THE paragon of Tech classes. Not even the masterful Quarian Engineer could stand up to him. First, you'd want to start with a unique power that none of the other Multiplater classes have. So I'd pick either Mordin's old Loyalty Ability Neural Shock, or maybe even the Defense Matrix from EDI, or the Sabotage ability to mess with your enemy's weapons if you didn't want to bring up a whole new one. Any of those would really set the tone of how to play the class. Next you'd obviously want Shield Drain, it's a solid ability that works like Overload but with a cherry on top. After that, I'd suggest a good strong ability like Incinerate to hit armor really hard when you needed it.
Those are just a few ideas I had. Anybody else have any input?