Fiona Apple -- Worn Thin

Fiona Apple has had a really rough week ... and it shows.

As we first reported, the 35-year-old singer was busted for weed and hash possession in Texas last week, then landed herself in a feud with the arresting officers, blasting 4 of the cops at a concert this weekend for allegedly mistreating her during the arrest.

One officer at the Hudspeth County Sheriff's Dept. responded to the attack, telling Apple to "just shut up and sing."

But it all appears to be taking a toll -- because the 35-year-old "Criminal" singer was spotted heading to her House of Blues concert in New Orleans last night looking gaunt and rail thin.

Touring can be brutal.