I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of all non-positively curved metrics on a given manifold (broad curvature assumption), or maybe generic properties of metrics on a surface (broad dimension assumption). It is primarily for my own general knowledge, but may be it could help others as well to have a handy list lying around. As examples of what could be possible answers, let me begin the list with two generic properties that come to mind immediately:
Generic simplicity of spectrum and generic properties of eigenfunctions: heuristically, the eigenvalues of the Laplacian for the generic metric on a compact manifold are non-repeated. Also, the eigenfunctions are generically Morse and have no nodal critical points (Uhlenbeck and Albert).
The generic Riemannian metric on a smooth manifold is real analytic. This can either be realized by local approximation in charts, or by applying the Ricci flow and using a result of Bando (mr=888130).
Please add your own favorite generic property. Thanks!
Edit: As commented below, the word "generic" needs clarification. I am using it to refer to properties that are too strong for just any given metric to have, but if you perturb the metric slightly, you can produce a metric that now has the said property.