Steiner
Kurita
Davion
Liao
Marik
Rasalhague
Mechs:
Firstly, a note on the model codes. As a general rule, mechs with designators ending in D are Davion variants, K are Kurita, L is Liao, S is Steiner, and M Marik. This is NOT a hard and fast rule. The JR7-D Jenner is an iconic Kurita light mech, the AS7-D is not a Davion specific mech either, the COM-1D and 2D are actually Steiner mechs, etc.
Secondly, the Project Phoenix mechs (P) variants (LCT-1V, SHD-2H, TDR-5S, BLR-1G, WVR-6R, and GRF-1N) are common throughout the Inner Sphere. Many of the other variants are house specific, however.
A last point before notes on individual mechs, remember that trade and battlefield salvage can cause mechs to appear in small numbers in other nations' militaries. The Federated Commonwealth treaty allows for easy trade between Houses Davion and Steiner; a similar treaty exists between Liao, Kurita, and Marik; Davions salvage Kurita Jenners, Kuritas salvage Davion BLR-1Ds, Rasalhague's using both Steiner and Kurita mechs, and with the clan invasion House Marik sells war material to almost everyone. You can justify nearly any Spheroid mech being fielded by any Inner Sphere faction, so don't sweat it if your favorite mech is not normally fielded by your faction of choice.
Light Mechs
Medium Mechs
Heavy Mechs
Assault Mechs