Via Robert Farley, Raoul Vega comments on starship design and battle tactics:
The Imperial Star Destroyer of the Star Wars universe is a hybrid battleship/aircraft carrier, capable (according to Wikipedia) of carrying 72 TIE fighters plus auxiliaries, but also capable of fighting it out ship-to-ship. I’m not sure that we ever get a really clear sense of which kind of fighting the Imperial Fleet prefers, since the Fleet does nothing during the Battle of Endor. A battlestar in both variants of Battlestar Galactica is also a hybrid. The original Galactica carried 150 fighters, while the ships from the newer series seem to carry about 40 or so (more on Pegasus, less on Galactica. Again, the BSG ships in both series engage in direct combat against Cylon Baseships, which in the second series in particular seem to resemble aircraft carriers much more than battleships.
All of this makes me wonder why the ships in Star Trek are so clearly cruisers and battleships, rather than aircraft carriers. As far as I can tell, no race in the series employs vessels that act as motherships to large numbers of fighters. Indeed, I suspect that the Enterprise (in all of its variants) would run into considerable difficulties if it were attacked by a swarm of small craft. Its phaser and torpedo banks seem designed to fight large ships, rather than to provide close defense against small craft. The same appears to be true of Klingon, Romulan, and other ships.
Clearly this is a question that calls for a made-up answer. So what I would say is that most likely in the Star Trek universe it’s not technologically feasible to equip a craft smaller than a Defiant-class starship with deflector shields. You could attribute that to the physics of the deflector fields themselves, or the need for a large power supply, or what have you. Either way, the upshot is that piloting a small craft in battle would be tantamount to suicide. At the same time, it’s probably also the case that it’s not feasible to build warp engines powerful enough to move a craft large enough to carry Defiant-class starships. Consequently, no aircraft carriers.
UPDATE: Looks like shuttlecraft do have shields but maybe they’re weak shields to protect against little rocks and such and not powerful enough to cope with weapons. Also, that link, which many have sent me, is from an “expanded universe” wiki so I don’t even consider it definitive. Memory Alpha doesn’t say anything about shields on shuttles or even runabouts.