Not module trays but I find these make my transplanting easy and I rarely damage them (pot or plantlet) http://www.inthegardenshed.co.uk/acatalog/Pots.html
I do prop the small ones up in poundland cat litter trays (purchased some years ago). I just swish the emptied pots in a bucket of water to clean them and store them in pots/boxes. The larger pots do need a few rinses to get them clean.