For basically any equivalence relation that I've encountered in my mathematical studies, the only problematic part (if at all) was proving transitivity. Is there a "real-world" (i.e. appearing in some field of mathematics) example of an equivalence relation where the reverse is the case? That is, an equivalence relation such that reflexivity and/or symmetry is not easy to see (and preferably one where transitivity is trivial on the other hand).

One such example can be found at the end of this answer, but it seems more like an illustration than something one actually encounters in one's studies.