So... What sort of identifying/decorative function could we assign to these patches, if they aren't related to the line of work of the character, or his or her rank, or whatnot?
Three symbols for three shifts, perhaps? But it would be a bit weird that we never actually see a shift consisting predominantly of those wearing the appropriate patch.
Three symbols for three pay grades (within each rank)? Scotty might have gotten a raise for the Memory Alpha IT work that was outside his usual field of proficiency...
Three symbols for level of excellence (within each rank), without practical impact on pay or the like? Scotty might be doing better on some weeks than others, but never quite reaches "star grade". It's much easier for those command types who don't have to master actual skills such as impulse engine repair.
Three symbols for time in rank? Scotty would be an old dog at LtCmdr with the highest possible symbol, the sphere, while Kirk would still be a youngster at Captain rank. But when seniority issues arise, even a difference of weeks is of significance, so three levels wouldn't help much.