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Polyomino squares

In a thread appeared in Nov. 2009 in the newsgroup it.hobby.enigmi, we searched for ways to cover squares with polyominoes. Here I collect some results on the 3 following problems.

For each kind of polyomino (say, for the pentominoes) we searched for the largest square that can be covered by that polyominoes. Livio Zucca found an incredible 210210 square made of 4410 decominoes. See the results!
We looked for square that can be formed by complete series of polyominoes (possibly disregarding those with holes). See the results!
I have also searched for square frames that can be filled with complete series of polyominoes. See the results!

Clearly, it is possible to extend the search in several directions....