One can define singular homology directly for an orbifold, mimicking the standard constructions, but I don't know if this has been written up. The alternative is to take a space that is homotopy equivalent to your orbifold, and take the homology of that. This has been worked out by Behrang Noohi in http://arxiv.org/abs/0808.3799.

If $G$ is a group, then the homology of the classifying orbifold of $G$ (the stack quotient of a point by the trivial action of $G$) is the group homology of $G$; so you see that it does not vanish above the dimension of the orbifold, which is 0, except in the trivial case.

One can define singular homology directly for an orbifold, mimicking the standard constructions, but I don't know if this has been written up. The alternative is to take a space that is homotopy equivalent to your orbifold, and take the homology of that. This has been worked out by Behrang Noohi in https://arxiv.org/abs/0808.3799.

If $G$ is a group, then the homology of the classifying orbifold of $G$ (the stack quotient of a point by the trivial action of $G$) is the group homology of $G$; so you see that it does not vanish above the dimension of the orbifold, which is 0, except in the trivial case.