Or any vessel in Starfleet, really. You never see Riker, Spock, Janeway, or Dax break out the old Dirt Devil and can of Pledge and I can't imagine that the ships have some kind of housekeeping department (hmm, would those be red/gold shirts?). Starbases and space stations would, and maybe the Galaxy and Sovereign classes due to their size, but not the smaller Mirandas, Excelsiors, and such. A possibility that occurs to me is maybe they clean using the ship's transporters? Maybe the main computer is tasked with scanning the ship for dust and dirt with the internal sensors, then beams the stuff into space. A more likely possibility is that the air filtration system on Starfleet vessels are just so damn good, that nothing ever gets dusty. Man, I would kill for that, especially when it comes to allergies.
Heh, the mental image of Picard vacuuming is priceless, though.