What constitute "known icing conditions" and when can you fly VFR while they exist?
The short answer is that it's up to you as PIC to make that determination; the FAA has no clear definition apart from what a "reasonable and prudent" pilot would do.

The FAA issued a legal interpretation on exactly this question in 2009, here's the closest thing to a definition that they give (my emphasis throughout):


  "Known icing conditions" involve instead circumstances where a
  reasonable pilot would expect a substantial likelihood of ice
  formation on the aircraft based upon all information available to that
  pilot. While "known icing conditions" are not defined by regulation,
  the term has been used in legal proceedings involving violations of
  FAA safety regulations that relate to in- flight icing. The National
  Transportation Safety Board (NTSB) has held on a number of occasions
  that known icing conditions exist when a pilot knows or reasonably
  should know about weather reports in which icing conditions are
  reported or forecast


They go on to say that the PIC is responsible for determining the likelihood of icing conditions:


  Any assessment of known icing conditions is necessarily fact-specific.
  Permutations on the type, combination, and strength of meteorological
  elements that signify or negate the presence of known icing conditions
  are too numerous to describe exhaustively in this letter. Whether a
  pilot has operated into known icing conditions contrary to any
  limitation will depend upon the total information available to the
  pilot, and his or her proper analysis of that information in
  evaluating the risk of encountering known icing conditions during a
  particular operation. The pilot should consider factors such as the
  route of flight, flight altitude, and time of flight when making such
  an evaluation


Note that even visible moisture and freezing temps aren't automatically considered to be known icing conditions:


  The FAA does not necessarily consider the mere presence of clouds
  (which may only contain ice crystals) or other forms of visible
  moisture at temperatures at or below freezing to be conducive to the
  formation of known ice or to constitute known icing conditions.


The specific regulations mentioned in the interpretation are 14 CFR 91.9 (compliance with the operating manual), 91.13 (no careless or reckless operation) and 91.103 (become familiar with all information relevant to the flight). Those regulations would be the basis for any hypothetical enforcement action:


  If the composite information indicates to a reasonable and prudent
  pilot that he or she will be operating the aircraft under conditions
  that will cause ice to adhere to the aircraft along the proposed route
  and altitude of flight, then known icing conditions likely exist. If
  the pilot operates the aircraft in known icing conditions contrary to
  the requirements of § 91.9(a), the FAA may take enforcement action.


So the bottom line is: you're PIC and it's your decision whether it's safe to fly or not because there are far too many variables for anyone else to decide. But personally I think it would be difficult for any pilot to say that it was "reasonable and prudent" to fly a non-FIKI aircraft in an area where forecasts, AIRMETs or PIREPS indicated icing.