In algebraic geometry, it is a sad fact of life that pushforward doesn't preserve being a coherent sheaf; for example, the pushforward by the complement of a divisor of the structure sheaf (or more generally a line bundle) has essentially no hope of being again coherent. On the other hand, on a smooth variety, if I pull out a closed subset of codimension $\geq 2$, then the pushforward of the structure sheaf will be be the structure sheaf of the whole thing, essentially by Hartog's theorem.

Now, this is not true for singular varieties; you can have an affine inclusion of varieties where the complement has codimension greater than 1, like removing the singular point from singular plane quadric.

I'm generally curious about when "not too bad" singularities can avoid this problem; I'm particularly interested in whether this works for removing the singular locus of a terminal variety, but would be interested to hear other results along the same lines.