For most crystalline solids (in which the atoms/ions/molecules are packed tightly), we expect the density to fall on melting. This is somewhat easy to see for elemental metals. For example, the density of fcc Al falls from 2.7 g/cc to 2.375, and that of fcc Cu from 8.96 to 8.02 g/cc on melting.
This trend is seen in bcc metals also: the density of Fe drops from 7.874 to 6.98 g/cc, and that of Cr from 7.19 to 6.3 g/cc on melting.
The ice-water transformation is a famous counterexample to this expectation; the density increases from 0.917 g/cc in ice to 1.0 g/cc in water.
From a point of view of packing, we know that the diamond cubic structure has a low packing factor of just only 0.34 (against 0.74 for fcc and 0.68 for bcc structures with spherical atoms), and therefore, we may expect the density of liquid to be higher than that of the crystalline solid for these elements. It turns out that this is indeed the case:
The density of Si rises from 2.329 to 2.57 g/cc on melting; so does that of Ge, from 5.323 to 5.6 g/cc.
In all these cases, the drop/rise in density is typically in the range of 5-15 percent or so.
[Note: All the above data on densities are from Wikipedia entries for the respective elements. The density of solid is for room temperature, while that of liquids is at the melting temperature.]
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Update (21 January 2015): Here’s a great demo of what a 9% volume expansion (~3 % linear expansion) due to freezing can do to water pipes: