Many existence proofs which exploit the idea of Baire category.
For instance, existence of a metrically transitive automorphism of the closed unit square was first obtained by the category method (see, e.g., see "Measure-preserving homeomorphisms and metrical transitivity" by Oxtoby and Ulam) . Another classical example is due to Banach who proved that every function from a residual subset of $C[0,1]$ is nowhere differentiable.
A nice and elementary book by Oxtoby where he discusses this these and many other applications of the category method)method.
