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It is worth noting some very interesting cases when the answer is yes. An amazing result by Miyata states that if $R$ is Noetherian and commutative, $M,N$ are finitely generated and $E \cong M\oplus …

This example also suggests that all ideals in $R$ have to be principal, otherwise similar counter-examples can be found. So you naturally gets to principal ideal rings. … You can replace "free" by "locally free" and play the same game, it will naturally leads you to all sort of interesting things worth learning about commutative rings, for examples projective modules or …

Take a normal non-CM domain $B$ over some field of characteristic $0$ (see this question for some concrete examples). Find a Noether normalization $A \subset B$ (Macaulay 2 can do it for you). …

You don't get more algebraic geometry than understanding the equations defining a (affine or projective) variety, and there are still tons of questions being actively studied. For example: 1) Google …

2) To weed out examples like above, perhaps more relevant than indecomposability or projectivity is to require $M$ to be torsion-free, so $M$ injects into $M\otimes Q(R)$ ($Q(R)$ is the total ring of … This perhaps explains why the easy counter-examples by me and Tom Goodwillie exist: $M$ can locally have a constant number of generators, but it generally does not mean $M$ is locally free. …

I would like to point out Teissier's Oberwolfach report about valuations and resolutions of singularity in positive characteristics. Also, more related to your comment here ,Cutkosky-Tessier has a r …