many linux distros will let you do this, but it is a bad idea.
HDDs transfer speed at aprox 80-120MB/s, flash drives (assuming you have a good one that can saturate the USB port) will cap out at 20MB/s. Most budget flash drives only have a read of about 10-15MB/s, and write at 2-3MB/s which would be painfully slow to run an OS on... but I suppose it is better than nothing.
Also, you use to need a bootable flash drive, as well as a mobo that supported booting off of USB, but I am not sure if this is true any longer.