The following uses for curium are gathered from a number of sources as well as from anecdotal comments. I'd be delighted to receive corrections as well as additional referenced uses (please use the feedback mechanism to add uses).
Curium is radioactive and only available in small amounts. Its uses are therefore rather specialist. For instance, curium was used as a thermoelectric power source and curium as 244Cm was the alpha particle source for the Alpha Proton X-Ray Spectrometer on the Mars Rover. See NASA Quest for more details, and where it is stated that:
"Week of August 25th
The mechanics then opened up the Lander and took the petals off. We are doing this so that we can install fresh batteries for launch. We also have to put in the Radio-isotope Heater Units (RHUs) on the Rover. These are devices containing a very small amount of Plutonium that give off heat to keep the Rover warm. We also installed a small amount of Radioactive Curium in the Alpha-Proton X-Ray Spectrometer (APXS) instrument on the Rover. This instrument uses the radioactivity in the Curium to give off alpha particles. When placed against a rock, these particles will hit the molecules in the rock, which in turn will release x-rays, Protons, and other alpha particles. The instrument looks at all three of these and determines what elements are in the rock."
and also explore Mars Exploration Rover Mission where it is stated that:
"The Alpha Particle X-ray Spectrometer (APXS) works by exposing martian materials to energetic alpha particles and X-rays from curium-244, and then measuring the energy spectra of backscattered alpha and X-radiation. The instrument is conceptually similar to but an updated and more capable version of the APXS instrument that was used on the Mars Pathfinder Sojourner rover.