Mantle is primarily a noun meaning (1) a loose, sleeveless coat or cloack, (2) something that covers, (3) the layer of the Earth between the crust and the core, and (4) the cerebral cortex. It also has a verb definition—to cover as with a mantle. Mantel is easy; it means (1) an ornamental protruding around a fireplace, and (2) a shelf above a fireplace. It has no other definitions.
Mantle is the correct spelling in the various forms of the phrases under the mantle and take/pick up/put on the mantle. The mantle in these expressions is a metaphorical cloak (see definition 1 above).
Examples
Confusion of the two words is rife on the web, but we’ll use positive examples. These writers use mantle well:
… but if you don the mantle of Family Values, you get pinned with family values. [comment on Washington Post]
… a nighttime snowfall blanketed my grimy neighborhood with a mantle of glistening white. [comment on Wired]
… he has taken to publicly saluting school superintendents like Dr. Brown who he deems to have taken up the mantle of shared sacrifice. [New York Times]
And these writers use mantel correctly:
Located on the top of Liberty Hill, the home … features an open living room and dining room with a fireplace set in a marble mantel. [San Francisco Chronicle]
The resort’s furniture, the wine cellar and the carved mantel over the fireplace in the bar were made on-site in his woodworking shop. [Montreal Gazette]
Your opinions that keeping ashes in an urn on the mantel seemed “creepy,” and scattering them in public places “slightly ghoulish,” were very strong. [letter to Daily Pilot]

