2 Added the proper hypothesis

Well, if you have a finite flat morphism as Matthew Morrow says above.

Also, this may or may not be relevant eventually, but with regards to analogues of (3) with the higher direct images (and in higher relative dimension, ie non-finite morphisms), you might also want to check out Steenbrink's paper (and Du Bois's earlier paper).

http://www.numdam.org/item?id=CM_1980__42_3_315_0

http://www.numdam.org/item?id=BSMF_1981_109_41_0

See in particular Theorem 1 (and Theorem 4.6).

It says that if $f : X \to Y$ is flat (EDIT: and proper) and the fibers have nice enough singularities, then $R^i f_* O_X$ is locally free for all $i$. There's also a recent preprint on the arXiv of Kollar and Kovacs on Du Bois singularities which deals with some things related to this at the end, see:

http://front.math.ucdavis.edu/0902.0648

1

Well, if you have a finite flat morphism as Matthew Morrow says above.

Also, this may or may not be relevant eventually, but with regards to analogues of (3) with the higher direct images (and in higher relative dimension, ie non-finite morphisms), you might also want to check out Steenbrink's paper (and Du Bois's earlier paper).

http://www.numdam.org/item?id=CM_1980__42_3_315_0

http://www.numdam.org/item?id=BSMF_1981_109_41_0

See in particular Theorem 1 (and Theorem 4.6).

It says that if $f : X \to Y$ is flat and the fibers have nice enough singularities, then $R^i f_* O_X$ is locally free for all $i$. There's also a recent preprint on the arXiv of Kollar and Kovacs on Du Bois singularities which deals with some things related to this at the end, see:

http://front.math.ucdavis.edu/0902.0648