Sharpness, or perceived sharpness, is influenced by a number of factors, but the effects of the sensor on that does depend a bit on some details around them. Typically, a full frame sensor has larger photo sites than an APS-C sensor which are, in turn, larger than a point and shoot. So, if we use this as the basis for discussion, understanding that there are exceptions to the "typical" case I described, then:
The larger pixel size provides some advantage in noise control which can assist in perceived sharpness (less detail loss), but it's bigger advantage is that it takes a smaller aperture to become diffraction limited as the increasing size of the airy disk can be contained inside the photo site for longer. For example, a Canon EOS 1D starts to become defraction limited at f/16 while a Nikon D70 hits it at f/11. Now, bear in mind that the physical dimensions of the sensor don't specifically effect this, it's simply a case that usually, though not always, the bigger sensor has bigger photo sites in current cameras. See this Cambridge in Colour article for much more detail.
Another factor in sharpness is contrast. The bigger sensors usually, though not always, have greater dynamic range and that usually leads to better contrast. If the contrast is better, the sharpness is better. Of course, this depends on the subject matter and if there is little to contrast in that, it won't really make a lot of difference what camera you use.
Now, on the downside, is that smaller pixel sizes may capture more detail at the same focal length, in effect, magnifying. This is where the crop factor, or focal length multiplier, comes into play. A 100mm lens on an APS-C (1.5) crop factor has the same angle of view as a 150mm lens on the full frame. Within that angle of view the APS-C, if not diffraction limited, will have captured more pixels worth of data than the full frame which gives it more detail and the potential to be sharper for the same region of the image assuming you shot using the 100mm lens on both. Which, by the way, is sometimes a disadvantage if you want shallow depth of field. It's one of the reasons that many struggle to get nice bokeh out of a point and shoot (though it can be done).
However, I started this reponse with an assumption of the "typical" condition of sensors, but the technology around this is improving dramatically and constantly. Sony (which generally supplies Nikon and Pentax) and Canon are making great strides in sensor technology and producing higher density APS-C sensors with improved dynamic range and substantially less noise. For example, cameras such as the Pentax K-x (Sony sensor) are getting rave reviews for their low-light noise control being in the same ballpark as a full frame sensors. So, while a full frame camera may some advantages in this area today, I don't think that's necessarily going to remain a constant truism. Now, these are "pro" cameras and will have a number of other features to go with them that others won't, so those may be compelling anyways.
On a side note, you excluded medium format digital, but they have one additional advantage over their full frame cousins from 35mm: no anti-aliasing filter. The filter helps with moire, but also introduces some blur. Users of this format would rather deal with the moire in post processing in order to get maximum sharpness on the sensor.
