  We study the distribution of squares in a Piatetski-Shapiro sequence
$\left(\lfloor n^c\rfloor\right)_{n\in\mathbb N}$ with $c>1$ and
$c\not\in\mathbb N$. We also study more general equations $\lfloor{n^c}\rfloor
= sm^2$, $n,m\in \mathbb N$, $1\le n \le N$ for an integer $s$ and obtain
several bounds on the number of solutions for a fixed $s$ and on average over
$s$ in an interval. These results are based on various techniques chosen
depending on the range of the parameters.
