  We consider an analogue of well-known Casson knot invariant for knotoids. We
start with a direct analogue of the classical construction which gives two
different integer-valued knotoid invariants and then focus on its homology
extension. Value of the extension is a formal sum of subgroups of the first
homology group $H_1(\Sigma)$ where $\Sigma$ is an oriented surface with (maybe)
non-empty boundary in which knotoid diagrams lie. To make the extension
informative for spherical knotoids it is sufficient to transform an initial
knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing
small disks around its endpoints. As an application of the invariants we prove
two theorems: a sharp lower bound of the crossing number of a knotoid (the
estimate differs from its prototype for classical knots proved by M.Polyak and
O.Viro in 2001) and a sufficient condition for a knotoid in $S^2$ to be a
proper knotoid (or pure knotoid with respect to Turaev's terminology). Finally
we give a table containing values of our invariants computed for all spherical
prime proper knotoids having diagrams with at most $5$ crossings.
