Not sure how old your children are, etc.; however, I had to recently make the same decision for our newborn son.
First, I'll try to answer your question, then I'll give you a personal recommendation which may/may not fit.
Your question: I bought my wife the Canon SD300 (4 MP) for Valentines day. The new SD500 is reportedly a real nice machine, as well. The SD300's pictures are very good; however, I do detect some noise even at relatively low ISO compared to my dSLR. Of course, such a comparison is probably unfair. The SD300 gives a small degree of control; however, it's doesn't provide any true manual control - it's a good P&S camera. The shutter lag (as w/most P&S) bugs me, but I don't know if you can get around that issue as the autofocus motors, etc. may all be limiting factors in such a small package. I've heard good things about the Casio Exilms (sp?); however, I have no firsthand knowledge. The websites listed by others are excellent resources to do research and view sample photos.
Personal recommendation: if you enjoy photo-/videography as a hobby, then you might be better off going with a dSLR & mini DV camcorder. As an example, I was going to get a simple P&S digicam; however, I was disappointed with the quality of the P&S pics & the shutter lag. I picked-up separate still & video cams for the following reasons:
1. dSLR: much better quality & flexibility (larger CCD/CMOS = less noise) and no start-up or shutter lag. These items were important because
- P&S CCD/CMOS are much smaller than those on dSLRs = more noise and problematic enlargements on P&S pics (if you plan on enlarging.
- Different cam mfg bias their CCD/CMOS for oversaturation/warmth to make the pics look "better" - I wanted a true representation and dSLRs tend to be more accurate from a color/tonality perspective.
- Shutter/start-up lag: the P&S I tried all had bad shutter lag - to the point I'd be recomposing my next shot before the first shutter had been released. Kids/children are fast-moving objects (or at least their expressions are) - I didn't want to miss critical moments, so minimal shutter lag was important.
2. mini DV: the quality of using a still digicam for movies (or a mini DV for stills) results in compromises (at least at this point in their development, based upon my research). Therefore, I bought a separate mini DV camcorder for movie purposes. I'm actually pleasantly surprised by the high quality mini DV can create these days - it exceeded my expectations. While I've captured a bit of DV, I can't say I've used FCE yet to edit and burn DVDs...but I hope to soon.
I might be suffering from "new parentitis" - trying to capture every moment for posterity (I hear the disease wanes with add'l children); however, I haven't regretted going separate dSLR & mini DV...albeit we still use a P&S when lugging a dSLR isn't practical...
At the end of the day, though, I've seen (and read) good things about the Canon, Nikon and Casio P&S. I have direct experience with the Canon SD300 (and have tried the S70 & S60 (discontinued?)) and have been pleased with their results.
Good luck and PM me if I can answer any questions.
Frank
edit: An idea if you want to compare cams - see what type of card your top choices use and go to a photo shop and take some test shots with your own card...then take the card home and compare side-by-side.