Supposing I owned a supersonic aircraft eg a MiG-21 and wanted to fly the jet out into international waters for a supersonic speed run. Are there any particular ATC procedures for doing so?
As a non-commercial civilian, the high seas (international waters) do not exempt you from the local governing rules: the civilian FIR boundaries meet over water, and there is no free-play gap in between. This goes as far back as 1948:
Flight over the high seas. It should be noted that the Council resolved, in adopting Annex 2 in April 1948 and Amendment 1 to the said Annex in November 1951, that the Annex constitutes Rules relating to the flight and manoeuvre of aircraft within the meaning of Article 12 of the Convention. Over the high seas, therefore, these rules apply without exception.
If your plan is to fly VFR and not file an IFR plan with the appropriate ATS authority, then you can't fly supersonic (unless authorized):
4.4 Unless authorized by the appropriate ATS authority, VFR flights shall not be operated:
a) above FL 200;
b) at transonic and supersonic speeds.
Reference: ICAO Annex 2 (Rules of the Air).
Note: Consult the local AIP for listed deviations from Annex 2 if the country of operation is an ICAO member.
In the US? You'd be talking to ATC to clear the ADIZ around the US coast. Then keep talking to them to make sure you aren't going to fly into any slower transoceanic traffic. And then to get clearance back into US controlled airspace.