  A standard tool for classifying the complexity of equivalence relations on
$\omega$ is provided by computable reducibility. This reducibility gives rise
to a rich degree structure. The paper studies equivalence relations, which
induce minimal degrees with respect to computable reducibility. Let $\Gamma$ be
one of the following classes: $\Sigma^0_{\alpha}$, $\Pi^0_{\alpha}$,
$\Sigma^1_n$, or $\Pi^1_n$, where $\alpha \geq 2$ is a computable ordinal and
$n$ is a non-zero natural number. We prove that there are infinitely many
pairwise incomparable minimal equivalence relations that are properly in
$\Gamma$.
