Monadic predicate logic (where all relation symbols are unary, and there are no function symbols).

The Bernays-Schönfinkel class of formulas (whose prenex form has the prefix $\exists z_1 \dots \exists z_k \forall y_1 \dots \forall y_m$ and contains no function symbols).

The relational fragment containing only a single binary relation symbol (and no function symbols or equality).

First-order logic restricted to using at most two distinct variables ($x$ and $y$).