Which SAT problems are easy?
I guess you know the classical result of Schaefer from STOC'78, but just in case.

10.1145/800133.804350

Schaefer proved that if SAT is parametrised by a set of relations allowed in any instance, then there are only 6 tractable cases: 2-SAT (i.e. every clause is binary), Horn-SAT, dual-Horn-SAT, affine-SAT (solutions to linear equations in GF(2)), 0-valid (relations satisfied by the all-0 assignment) and 1-valid (relations satisfied by the all-1 assignment).