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][1]

[http://www.numdam.org/item?id=BSMF_1981__109__41_0][2]

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:

[https://arxiv.org/abs/0902.0648][3]


  [1]: http://www.numdam.org/item?id=CM_1980__42_3_315_0
  [2]: http://www.numdam.org/item?id=BSMF_1981__109__41_0
  [3]: https://arxiv.org/abs/0902.0648