  The combinatorics of squares in a word depends on how the equivalence of
halves of the square is defined. We consider Abelian squares, parameterized
squares, and order-preserving squares. The word $uv$ is an Abelian
(parameterized, order-preserving) square if $u$ and $v$ are equivalent in the
Abelian (parameterized, order-preserving) sense. The maximum number of ordinary
squares in a word is known to be asymptotically linear, but the exact bound is
still investigated. We present several results on the maximum number of
distinct squares for nonstandard subword equivalence relations. Let
$\mathit{SQ}_{\mathrm{Abel}}(n,\sigma)$ and
$\mathit{SQ}'_{\mathrm{Abel}}(n,\sigma)$ denote the maximum number of Abelian
squares in a word of length $n$ over an alphabet of size $\sigma$, which are
distinct as words and which are nonequivalent in the Abelian sense,
respectively. For $\sigma\ge 2$ we prove that
$\mathit{SQ}_{\mathrm{Abel}}(n,\sigma)=\Theta(n^2)$,
$\mathit{SQ}'_{\mathrm{Abel}}(n,\sigma)=\Omega(n^{3/2})$ and
$\mathit{SQ}'_{\mathrm{Abel}}(n,\sigma) = O(n^{11/6})$. We also give linear
bounds for parameterized and order-preserving squares for alphabets of constant
size: $\mathit{SQ}_{\mathrm{param}}(n,O(1))=\Theta(n)$,
$\mathit{SQ}_{\mathrm{op}}(n,O(1))=\Theta(n)$. The upper bounds have quadratic
dependence on the alphabet size for order-preserving squares and exponential
dependence for parameterized squares.
  As a side result we construct infinite words over the smallest alphabet which
avoid nontrivial order-preserving squares and nontrivial parameterized cubes
(nontrivial parameterized squares cannot be avoided in an infinite word).
