HX9V or ZS-20 or F770EXR?
I tried several similar cameras recently - I settled on the Panny ZS15; I find it a nice camera to use, and the images are fine for my purposes up to about ISO 800. According to some websites it's a little better than the ZS20 for low-light shooting, but has a little less zoom, no GPS, no touch screen and fewer pixels; all perfectly OK with me. Panasonic was first in the sub-genre of pocket super-zooms, and their experience shows - the IS and lens quality is spectacular compared to most others in the category. I have had no issues with the image quality until I hit ISO 1600, when fine detail starts to degrade. I really like Panasonic's approach to noise - they allow some luminance noise to preserve details better; and NO color blotching from over-aggressive chroma noise suppression.
There is a detailed thread with lots of samples in the Fuji forum regarding the F770 - http://forums.dpreview.com/forums/read.asp?forum=1012&message=40769063 - the OP clearly likes the camera. I own a Fuji F505 EXR (a precursor to the F770) and it's a delightful camera to use, very comfortable and intuitive. Image quality is sub-par, though - I gave it to my wife and she's had no concerns with it; as a snapshot camera it's very good. She's getting better results from it than I was; kind of humbling considering how long I've been at this.... According to the poster mentioned above, the F770 might have licked most of the serious problems that the previous cameras in the series were criticized for.
As for the Sony, I never tried it - I have a thing about Sony since I got rootkit-ed and lost important data in the ensuing panicked mess; I was a died-in-the-wool Sony fanboi with several Sony video cameras and other electronics until they did that to me....
The only other thing I'll mention is that you really should take a look at the Canon SX260; I fondled one recently at the store and it's really nice to hold and use, much improved from the (quite nasty, IMHO) SX230. It's likely got the best image quality of the bunch, according to early reports.
Good luck in your search, hope I've helped.