Transit surveys for Earths in the habitable zones of white dwarfs". The author, Eric Agol, makes a careful and thorough study of the potential characteristics of Earth-type planets orbiting close enough to white dwarf stars to meet the usual rudimentary criteria for habitability (i.e. liquid surface water).
White dwarfs, the remains of the stellar cores of roughly solar-mass stars, are tremendously compact objects supported by electron degeneracy pressure - the direct manifestation of quantum mechanical exclusion, close packed electrons don't like their wave-functions overlapping. As Agol points out, a typical white dwarf is about the same size as the Earth. This would result in a doozy of a planetary transit signature (white dwarf on, white dwarf off). He then goes on to figure out what the orbital configurations would need to be around white dwarfs of varying ages and temperatures for a planet to hit the "habitable" mark.
White dwarfs are low luminosity - they're just very small - so this zone is about 0.005 AU to about 0.02 AU for a range of parameters, and lasts for at least 3 billion years as the dwarfs cool off. Planets this close in to dwarfs will have orbital periods on the order of about 10 hours. So the odds of catching transits, which are going to be extremely deep as the planets block out most of the light, are really good. Rather neatly, since the white dwarfs are so dense, such close planets will be unable to raise a tidal bulge on the star and so their orbits are likely to remain stable over long timescales.
The catch is that these planets may or may not exist, and if they do they may have very uninteresting compositions. On its way to becoming a white dwarf a solar type star will inflate its outer atmosphere all the way out to about 1 AU. This is probably the ultimate fate of the Earth, to be engulfed by the star that has nurtured it for the previous 10 billion years. Even planets outside this puffed up stellar envelope may get destroyed as tidal effects perturb their orbits, some estimates suggest that even 3 AU is not a safe distance. Furthermore, as the star loses as much as 50% of its mass before ending up as a white dwarf the fundamental dynamics of any outer planets is changed. Orbits expand outwards and the mutual Hill radii, or range of influence of planets increases. This can lead to planet-planet scattering events that rearrange the entire planetary architecture.
Despite all this, as Agol points out, we do know that planetary bodies can exist even around neutron stars - the pulsar planets. Other observations also suggest disks of material, possibly containing planetary sized bodies, around stellar remnants. Just because one batch of planets gets destroyed doesn't mean that material can't get recycled into forming "new" planets. So there may be pathways for nature to rebuild or rearrange planets to put them right in the habitable zone of a white dwarf. Whether they would have compositions that include either water or young radioactive elements (vital for maintaining internal heat and hence geophysical activity) is rather a taller order to satisfy.
Nonetheless, if we've learned anything from exoplanetary science it's that nature is going to surprise us at every turn. Spotting worlds around white dwarfs would be immensely cool, er, temperate, and would undoubtedly yield a host of insights to the nature of planet formation in general - even if these are dry and inert lumps of hand-me-down material.