You can check out reviews on dpreview.com.
The D5100 is here and you can see it compared to the D3100.
http://www.dpreview.com/reviews/nikond5100/
The Canon mentioned is reviewed here:
http://www.dpreview.com/reviews/canoneos1100D/2
I didn't see a review there for the P510, but you can do some searches on the site for any of the cameras and see what others think.
In terms of continuous shooting for sports, it looks like the D5100 is 4 fps (frames per second) for continuous shoot, the D3100 is 3 fps, and the Canon is 2 -3 fps depending on which file format you are using. The P510, according to the Nikon website, looks higher. The other factor is how many pictures you can take in a row at that speed. That usually depends on how fast it can process and write the images to the memory card. For example, the P510 says it can to up to 5 shots at 7 frames a second. That means you couldn't do another burst of shots until it had finished with at least some of the previous ones. Depending on your needs, that may not matter.
I wouldn't expect you to notice the difference between a 12 megapixel or 16 megapixel camera unless you are doing poster size or something.
If you have a chance to play with the cameras in a store, I would do that. Practice trying to take pictures quickly with both and see if/when you notice a delay in continuous mode. Also check out other things that might influence your happiness with the camera. For example:
- How heavy does it feel? Does it feel bulky. Is the DSLR with the lenses on going to bug you with respect to size and carrying it around.
- How soon after turning it on can you take a picture?
- Is there a delay between the time you press the shutter and when the picture is actually taken. Point and Shoots cameras, particularly super cheap or old ones, are really bad in this category. Newer ones are less of an issue.
- What kind of optical zoom does it have? For a DSLR that will depend on the lenses, but you can check it out for the lenses that come with it. Please ignore any specs about digital zoom. Digital zoom can always be done after taking the picture and it's true for any camera.
- See if you can take a picture that gives you a feel for how well the flash works. I had a Point and Shoot once that took great outdoor pics, but all the indoor pics were terrible.
If the P510 does the continuous mode in a way that you like, then I suspect it might best fit your needs, especially if you won't be fiddling with any manual settings.
I have an older DSLR that I used a lot for sports and I suspect I will use again once my kids are old enough to participate in them. But I also have a small point and shoot that I prefer when out and about as it's much easier to carry. My DSLR gives better pics than my point and shoot, but it's also more work. So I will use it if I want special pictures, but for day to day, I use the point and shoot.