I've used a lot of point and shoot cameras over the years, and I'm always looking for a point and shoot, small camera with a great lens and decent control, i.e. manual mode or at least aperture control. I've also used some higher end camera bodies and some good lenses. What I've carried for the past few years in a Panasonic DMC-FX01, and I really like it.
Back in the film days, I had a lot of point and shoot cameras, the two best being a Yashica T4 and a Ricoh Gr1 (here's a good comparison of the two on photo.net) and I loved them both, but the Gr1 was a better camera, albeit for much more money. The T4 had a brilliant lens for little money, but allowed the photographer no control. The Gr1 a brilliant lens at a better focal lenth, 28mm, and manual control, for a lot of money. I made some good negatives with both.
That's the past, and there are a lot of great digital cameras out there now. The Panasonic DMC-FX01, despite having no manual control, just keeps taking great pictures for me. As with all cameras in this class, indoor photography using flash sucks, so I have another camera for that, but for a carry around, have with you all the time, get the shot camera, this thing is very good. Anyone can make good pictures with a camera like this.
Just use it right. For me that means:
Flash off indoors, mostly. Find a way to stabilize the camera. This was taken with no flash at around 1/4 of a second. I like this picture. I stabilized the camera on the table and used the self timer.
Hand held self portrait. This again with the DMC-FX01. Not bad for a hand held self portrait. We're outside, so flash is on. This photo is overexposed by about 1/2 a stop, but it's OK other than that. No one asked you about the goggles or the helmet.
Here I'm just banging away, and accepting the camera's exposure. It's fine, a decent compromise that works for the subject matter. Again, this is just amateur documentary photography and a photo like this works just fine for me.
If you are like me and you want a camera you can carry nearly all the time, but you want good results, the DMC-FX01 or something like it is wonderful. There are great posh point and shoots that offer more control if you want to spend a lot more money. But my point is that if you think even a little bit about how to use your cheaper point and shoot camera, you can get good results. You don't need an expensive rig to make some fine, useable photos.