Utah
They appear to be prohibited currently, but things are changing.
Permit is needed for commercial aquaculture:
http://ag.utah.gov/licensing/documents/aq_COR_app.pdf
It is unclear what is required for personal use.

Also see
Utah Aquaculture Association
Mark Leavitt / Esther Williams
3700 E. Glenwood Road
Richfield, UT 84701
Tel: (801) 836-2485; Fax: (801) 836-2489