Where is RC aircraft flying allowed in the US?
Update

Everything in the second section (my original answer) used to be true (well, except for the first statement), but doesn't consider that the FAA Modernization and Reform Act of 2012 made this regulatory.

The FAA has a webpage that discuss it in more detail , but here is an image that covers the basics:



There is also a much more detailed letter of interpretation from the FAA which discusses their ability to enforce the regulation and many other parts of 14 CFR 91 if a model aircraft endangers another aircraft, flies within 5 miles of an airport without receiving a clearance, flies into a restricted area, etc.



The short version is that it isn't regulated (yet).  For more details, read on.

FAA has an Unmanned Aircraft (UAS) Questions and Answers page on their website which addresses this:


  Do I need permission from the FAA to fly a UAS for recreation or as a hobby?
  
  There are two ways for recreational or hobby UAS fliers to operate in the National Airspace System in accordance with the law and/or FAA regulations.
  
  ...
  
  Option #1. Fly in accordance with the Special Rule for Model Aircraft (Public Law 112-95 Section 336). Under this rule, operators must:
  
  Fly for hobby or recreational purposes only
  
  Follow a community-based set of safety guidelines
  
  Fly the UAS within visual line-of-sight
  
  Give way to manned aircraft
  
  Provide prior notification to the airport and air traffic control tower, if one is present, when flying within 5 miles of an airport
  
  Fly UAS that weigh no more than 55 lbs. unless certified by a community-based organization
  
  Register the aircraft (UAS over 0.55 lbs. and less than 55 lbs. can be registered online at registermyuas.faa.gov; UAS 55 lbs. or greater must be registered through the FAA's paper-based process)


The other option is to obtain a pilot certificate and comply with part 107 UAS regulations.

The FAA has also published an Advisory Circular Model Aircraft Operating Standards, which states:


  1 - PURPOSE. This advisory circular (AC) provides guidance to persons operating Unmanned Aircraft (UA) for hobby or recreation purposes meeting the statutory definition of “model aircraft” contained in Section 336 of Public Law 112-95, the FAA Modernization and Reform Act of 2012. This AC describes means by which model aircraft may be operated safely in the National Airspace System (NAS). Nothing in this AC changes the requirement to comply with the statute or any applicable regulations.
  
  ...
  
  6 - MODEL AIRCRAFT OPERATIONS. 
  
  ...
  
  b. Model Aircraft Hazards in the NAS. While aero-modelers generally are concerned about safety and exercise good judgment when flying model aircraft for the hobby and recreational purposes for which they are intended, they may share the airspace in which manned aircraft are operating. Unmanned aircraft, including model aircraft, may pose a hazard to manned aircraft in flight and to persons and property on the surface if not operated safely. Model aircraft operations that endanger the safety of the National Airspace System, particularly careless or reckless operations or those that interfere with or fail to give way to any manned aircraft may be subject to FAA enforcement action.
  
  c. Determination of “Model Aircraft” Status. Whether a given unmanned aircraft operation may be considered a “model aircraft operation” is determined with reference to section 336 of Public Law 112-95:
  
  (1) The aircraft is flown strictly for hobby or recreational use;
  
  (2) The aircraft operates in accordance with a community-based set of safety guidelines and within the programming of a nationwide community-based organization (CBO);
  
  (3) The aircraft is limited to not more than 55 pounds, unless otherwise certified through a design, construction, inspection, flight test, and operational safety program administered by a CBO;
  
  (4) The aircraft operates in a manner that does not interfere with, and gives way to, any manned aircraft; and
  
  (5) When flown within 5 miles of an airport, the operator of the model aircraft provides the airport operator and the airport air traffic control tower (when an air traffic facility is located at the airport) with prior notice of the operation. Model aircraft operators flying from a permanent location within 5 miles of an airport should establish a mutually agreed upon operating procedure with the airport operator and the airport air traffic control tower (when an air traffic facility is located at the
  airport).
  
  d. Public Law 112-95 recognizes the authority of the Administrator to pursue enforcement action against persons operating model aircraft who endanger the safety of the National Airspace System. Accordingly, model aircraft operators must comply with any Temporary Flight Restrictions (TFR).
  
  ...
  
  Model aircraft must not operate in Prohibited Areas, Special Flight Rule Areas or, the Washington National Capital Region Flight Restricted Zone, without specific authorization.
  
  e. Model aircraft operators should follow best practices including limiting operations to 400 feet above ground level (AGL).
  
  f. All other operators and for additional information on Unmanned Aircraft Systems please visit: http://www.faa.gov/uas/ .


Note that this is "guidance" and recommended procedures that the FAA "encourages voluntary compliance with" in order to "reduce the potential for hazards and create a good neighbor environment with affected communities and airspace users."

That being said, any city/county/state could have rules on their books, but there is not national guidance for this.