Instinctively, I say ah-LEE-sha because I assume that's how everyone else will pronounce it, however, I much, much prefer ah-LEE-see-uh. BTW, I had never heard of the 'lish' or 'liss' pronunciations until I read those online.
I live in
Virginia and I've only known one my age and she spelled her name Aleasia. I wonder if her parents were going for the four-syllable pronunciation and spelled it phonetically. Didn't work since everyone ended up calling her
Lisa. Overall, I wouldn't say it's common where I'm from. I have seen it a time or two in my kids' yearbooks (usually spelled
Alicia or
Alisha). I suppose it could be considered dated by some. I've just never known many so it seems pretty fresh in my mind.
Re: pronunciation, I've wondered myself if
Alicia Silverstone influenced the public's awareness, but overall, I think not. The default way in the US seems to be ah-LEE-sha and that's the one I feel people will revert to. Still, the name isn't overly common and I think you could get away with any pronunciation. Among the three names you listed, I prefer
Alice,
Alicia and
Alyssa in that order.