Let $X$ be a 2n-dimensional manifold on which the $n$-dimensional compact torus $T$ acts in a locally standard manner. Let us consider the filtration of $X$ by torus invariant subsets $X_i$, where $X_i$ is the union of all $i$-dimensional orbits. My goal is to describe the structure of homological spectral sequence associated with this filtration. In the classical case, that is when $X$ is a quasitoric or toric manifold, this spectral sequence collapses at the second page. The same holds when all faces of the orbit space are acyclic. In more general situations the spectral sequence does not collapse at the second page. Nevertheless sometimes it can be described in full.