No - at least not in the sense of some shared data structure that different music editors were all capable to work with, as was the case with fx MOD files and derivatives.
C64 music is typically distributed as executables which contain both play routine and music data (cf. also this answer). Many different players and editors exist, some native, some for "cross composing". The music is played by the player software continuously changing the SID chip's parameters.
Composers, or groups, often wrote the play routines themselves and held on to them, and the particulars of a given implementation was often part of the artist's distinctive style. Also, play routines evolved a great deal, everyone trying to get an edge in sound capability, raster time usage or whatever trade-off of the system's constraints were relevant in the context.
There is, however, a pretty standardized meta data header - PSIDPSID - that represents the information needed to play the music from emulating music players or for generic front ends on the C64 itself. But this was something that was invented after the fact, when people started building collections and wanting to play the music on other systems as well (fx all the tracks you find in collections such as the HVSC are wrapped in this header).
There was not really any single "leader of the pack", but there were a few play routines used by popular and productive composers that you were likely to encounter, such as:
- Rob Hubbard's
- Martin Galwin's
- Matt Gray's
- 20CC's
- Vibrants'
And also those of some widely available music editors such as:
- Future Composer
- Music Assembler
- DMC