I think one example is given in [this][1] MO question of mine: a quartic in $\mathbb{P}^3$ with at worst Du Val singularities is a K3 surface (and similar statements for two types of complete intersections in higher-dimensional projective spaces). 

Using the excellent answer and comments I was able to piece together a proof, but I could not locate one in the literature, whereas of course the result was "well-known to experts" (to such an extent that I even felt embarrassed for asking about the proof in the first place).

[1]: https://mathoverflow.net/questions/200129/singular-models-of-k3-surfaces