Shelf is always a noun, and shelve is always a verb. When you shelve something, you put it on a shelf. The main definitions of the noun are (1) a horizontal surface supported by a vertical structure and used to store things, (2) a layer of ice or bedrock, and (3) something that resembles a shelf. The verb, shelve, means (1) to put [something] on a shelf, (2) to set aside, or (3) to slope gradually.
The plural of shelf is shelves. Shelve is inflected shelved, shelving, and shelves.
Examples
Shelf
One out of every eight boxes of cereal to leave the shelf in America carries the Cheerios name. [AP]
Her walk-in closet includes shelves all the way to the ceiling. [New York Times]
The environmental group says this includes seafood labelling, tracing fish from ocean to shelf and avoiding the sale of fish that are illegally caught. [CTV]
Shelve
Councillors in Bath have voted to shelve plans for a controversial park-and-ride at Bathampton Meadows. [BBC]
Credit cards have also been shelved until the end of the tournament. [Sydney Morning Herald]
A major American utility is shelving the nation’s most prominent effort to capture carbon dioxide. [New York Times]