As for most supported, that would be Blu-ray in both movie studios and tech manufacturers. Here's a list of the studios, those currently exclusive to one format have been emboldened. I don't think this list is entirely correct, but it gives you an idea.
HD-DVD - Universal, Warner Brothers, HBO, New Line, Paramount and Dreamworks.
Blu-ray - Sony Pictures (Columbia-Tristar), MGM, Fox, Disney, Lionsgate, Paramount, Dreamworks, Warner Brothers, HBO and New Line.
Both systems use all the same video and audio format, but Blu-ray has a higher capacity so there is a potential for less compression or more video. I'm gonna guess that most studios will use the same compressed source for both though as I doubt many will fill the disc space with their feature movie even on the smaller HD-DVD. So I doubt a difference in video quality would be noticable, but you might have more bonuses or less discs in special edition packs with Blu-ray. HD-DVD dual layer is 30GB, Blu-ray dual layer is 50GB. The HD-DVD camp have got a triple layer 45GB disc working, whereas the Blu-ray camp have a six layer 200GB disc working.
Currently Blu-ray discs will be more expensive to produce than HD-DVD, but if this generation lasts as long as DVD (unlikely) I think it is highly likely Blu-ray discs will end up the cheaper of the two.
One thing I'm excited about (after hiring so many unwatchable DVDs) is Blu-ray's new hard coating (TDK's Durabis) on the disc. Again more expense, but it is a lot better protection than previous optical discs. The media hype is 100 times the scratch resistence of DVD. It is also smudge and dust resistant and can be cleaned with a tissue (which of course scratched DVDs). When they announced the coating they gave each journalist a marker and steel wool and asked them to try damage the discs. The only guy that succeeded was one who pulled out his pocket knife! HD-DVD is essentially the same as DVD, uncoated Blu-ray is a lot worse than both (before Durabis was introduced Blu-ray discs were going to have to go in caddies like DVD-RAM).