Most Linux distributions, such as Ubuntu, are capable of installing directly to a USB drive as if it were just another hard drive, but some distributions treat USB flash drive as a slightly special case. DSL, for instance, has an installer specifically for creating a USB-drive installation of the OS. Puppy can also install itself directly to a flash drive through the built-in Universal Installer utility. However, the default options did not work in my case; I had to explicitly use the "SysLinux" option in its installer menu to make the flash drive bootable.
In all cases, a USB flash drive will be mounted and recognized as a virtual SCSI device, such as /dev/sda or /dev/sdb. If you're performing the installation on a system that already has a hard drive present, pay close attention to the devices listed in the partition manager, and make sure you're installing to the right device. Otherwise you might accidentally wipe the hard drive -- although the device size should be a big tipoff! Also make sure that the installer makes the target device bootable and writes a proper master boot record to the drive, although most of the time this should happen automatically.