Suppose $X$ is endowed with the trivial topology, e.g. $X$ and $\emptyset$ are the only open sets. For the sake of simplicity, I'll assume that $X$ is finite, $|X| = m$.
Now, the $n$-th module of the singular chain complex of $X$ should be a free $\mathbb{Z}$-module generated by the basis of every map from the n-simplex to $X$, since all those maps are continuous.
Looking back at the example of a single point space, the boundary operators could be described explicitly in terms of $n$ odd or even since the cardinality of the basis was always one. In this case, however, I fail to see what those maps are supposed to look like. I'd appreciate some hints on how to tackle this problem.