Continuity mistake: During the last fight with the Reavers, Inara is holding a blaster gun shaped like a bow and arrow in close up shots. In any shot that shows the whole crew, the blaster changes into an actual bow and arrow. The producers didn't like the bow so they actually enhanced it with CGI to make it bigger and "cooler", so those shots of the whole crew are probably shots where they forgot to add the CGI or assumed people wouldn't notice.
Revealing mistake: When the Operative is reviewing Mal's file, Mal's first name is initially spelled as "Malcom", in the next shot that has been corrected to "Malcolm".
Continuity mistake: Simon rescues River from the testing facility. He removes a needle from her forehead which leaves a spot of blood. He tries to wipe the blood and smears it. However, when the two leaves the room, the smear has reverted back into a dot.
Answer: She has several mental disorders, presumably she finds shoes confining or uncomfortable. It's notable that when she does have footwear it's almost always heavy calf high boots, instead of slippers or more comfortable shoes.
Grumpy Scot
To add; She likes to dance, and being barefoot it's easier to move around.
lionhead
Something I always wondered is when she goes on the heist barefoot at the beginning of the film, none of the other characters seem to notice or care that she's in bare feet. Even her brother, Simon, didn't try to get her to wear shoes.