I know there is controversy over the safety and utility of performing spins for training, but what are the regulations limiting them for training (solo and dual) or 'just for fun'?
There are at least three FARs that apply (maybe more?). First, "aerobatics" as defined in 91.303 seems to include spinning:
For the purposes of this section, aerobatic flight means an intentional maneuver involving an abrupt change in an aircraft's attitude, an abnormal attitude, or abnormal acceleration, not necessary for normal flight.
That means the general restrictions in 91.303 would apply:
No person may operate an aircraft in aerobatic flight—
(a) Over any congested area of a city, town, or settlement;
(b) Over an open air assembly of persons;
(c) Within the lateral boundaries of the surface areas of Class B, Class C, Class D, or Class E airspace designated for an airport;
(d) Within 4 nautical miles of the center line of any Federal airway;
(e) Below an altitude of 1,500 feet above the surface; or
(f) When flight visibility is less than 3 statute miles.
Second, spinning an aircraft not approved for spinning in the POH would be a violation of 91.9:
§91.9 Civil aircraft flight manual, marking, and placard requirements.
(a) Except as provided in paragraph (d) of this section, no person may operate a civil aircraft without complying with the operating limitations specified in the approved Airplane or Rotorcraft Flight Manual, markings, and placards, or as otherwise prescribed by the certificating authority of the country of registry.
Finally, 91.13 would also prohibit spinning a non-certificated aircraft because it's a 'catch-all' regulation:
§91.13 Careless or reckless operation.
(a) Aircraft operations for the purpose of air navigation. No person may operate an aircraft in a careless or reckless manner so as to endanger the life or property of another.
But spinning itself is absolutely legal and is required for a flight instructor rating as described in 61.183.
EDIT: there's also a requirement in 91.307 that if non-crewmembers are on board then parachutes are required for maneuvers with 'extreme' bank or pitch, unless the flight is a test or training flight:
(c) Unless each occupant of the aircraft is wearing an approved parachute, no pilot of a civil aircraft carrying any person (other than a crewmember) may execute any intentional maneuver that exceeds—
- A bank of 60 degrees relative to the horizon; or
- A nose-up or nose-down attitude of 30 degrees relative to the horizon.
(d) Paragraph (c) of this section does not apply to—
- Flight tests for pilot certification or rating; or
Spins and other flight maneuvers required by the regulations for any certificate or rating when given by—
(i) A certificated flight instructor; or
(ii) An airline transport pilot instructing in accordance with §61.67 of this chapter.