Have a look at
this perfect little post (I think this should be put in the FAQ somewhere) for a very concise description of how MP3 works.
When the data is converted back to the temporal domain (decoded), the result is an approximation of the original PCM values (ranging from -32768 to 32767 in the case of 16 bits precision). But since it is only an approximation, some sample values may exceed the scale: a sample that should decode to 32800 will be represented as 32767 (a deviation of this magnitude will not be audible, though). One way to avoid this is to allow for some headroom while decoding. The MAD software decoder does this, AFAIK, but I have never heard of hardware decoders with this capability. Foobar2000 takes another approach: it decodes to a 32 bit floating point format where the scale has no (practical) limitation. The "normal" scale goes from -1 to 1, but samples can exceed the scale without a problem. Clipping will occur only at the end of the processing pipeline, when the data is converted to the fixed-point format the soundcard requires (like the usual 16 bits) - unless you've been clever enough to use something like foobar's soft clipping or advanced limiter DSPs. They will scale parts of the signal in a sensible way so that it stays within the limits of the scale and doesn't clip.
Another possibility to avoid clipping MP3s is to use MP3gain, as indybrett has already pointed out. MP3gain decodes your MP3s and if it detects clipping, it can (losslessly) reduce the volume of the file to the point where no clipping occurs.
Now for audibility: I think the clipping introduced by an MP3 decoder should not be audible for most people (and certainly not when driving a car!). As has been pointed out many times before, many of the (pop/rock) CDs you can buy today have been clipped to death anyway due to bad mastering/production. The clipping may get worse (and audible), however, if the decoded signal is run through a DSP (like an equalizer).
If you want to know what clipping sounds like, get foobar2000 with the VU meter visualization component: whenever the top bar (labeled "Peak") turns red, foobar has detected clipping. You can cause more clipping by dragging the preamp sliders in Preferences/Playback to the right (note that for the changes to become effective playback has to be restarted).