My digital P&S has a range of fps burst rates: 2, 5, 10, 40, 60. Of course with the higher you go, there is a decreasing quality. Whichever, I have found that light is often a problem, and I must use a high ISO. My shots are mostly of rambunctious grandkids, and I found that the 60fps results in the least blur, but of course there is the quality price to pay. Thank you.
The burst rate should be based on the amount of action and how closely you need to be able to catch the shot. You should be able to adjust shutter speed to get less blur even if only doing 2 frames per second (or even just a single photo).
I would recommend using a shutter fast enough to stop the motion with enough light to prevent needing high ISO and enough burst to be able to catch the shot you are looking for without reducing the quality further than you need to.
It is not so much the burst rate but the shutter speed that makes your photo blurred or not. For normal walk ~1/125s will "freeze" the motion, while for "rambunctious" (I'll have to check the exact translation of that but :-D anyway) kids I would recommend at least 1/500s. This can indeed be made possible by increasing the ISO (sensitivity), but that also depends on the maximum aperture of the lens on your camera (for example f/2.8 or better will help of course - aperture or speed priority modes recommended...).
The higher fps rates will allow you to choose the best shot among a burst with more photos during the same timespan, for example without closed eyes or such defects. However, depending on the camera model, the focus may not be updated between the shots of the burst, resulting in increasingly blurred photos if the subject runs towards you (for example).
Practically I would recommend a rate that still maintains a good image quality (but it is difficult to give you a precise answer without any camera information). For reference, for sports and other rapid action, professional DSLRs with ~15fps seem sufficient. I am quite happy with ~7fps in 99% of the shooting circumstances I encounter.
One thing to take note of: with most P&S cameras with options such as you describe in your question the resolution of images taken at the higher frame rates is lower than the maximum resolution of the camera. If you are using full 100% magnification on your screen to look for blur in the images, you are magnifying the full resolution images at a much higher ratio than the lower resolution images. This will cause the same amount of blur to look worse because you are looking at it at higher magnification. If, for example, your monitor is 1920x1080 then the fast frame rate HD quality images will fill your screen without any magnification. A 12MP image, on the other hand will fill your screen 6 times over when magnified to 100%.
The 10 fps, 40fps, and 60fps second options are basically shooting video and saving each frame as an individual image. As you noted in your question, there is a price to pay in terms of quality. Even HD video is only about 2 megapixels per frame and is usually compressed more heavily than still images are. The biggest part of the compression is based on the parts of the scene that don't change from frame to frame, but there can also be heavier compression of each frame.
Which works best for you depends a lot on the intended use of the images. A 720x480 (or even lower resolution) image posted to facebook requires different resolution and quality than a 16x20 framed print. And of course there are many other uses for an image in between those extremes.
Which of the choices your camera offers will be the sweet spot for you will vary by the use of the photo. In general, I would use the fastest frame rate that allows for the full resolution of your sensor if that is fast enough to follow the action of your grandchildren. Even with the slower frame rates you can try to get your camera to use a faster shutter speed. If you can't control the shutter speed manually or in shutter priority mode try using the sports scene mode that many cameras offer.