To see rosette comps, your horse needs to have high enough skills in the primary skill of the competition in question (e.g. speed for barrel comps, stamina for XC comps, etc). At the moment, I believe roughly 7500 skills are necessary to see most rosette comps -- although this can vary between disciplines.

Keep in mind that most rosette comps are reserved for horses with more than 20 wins, so if your horse(s) don't have 20 wins yet, it may be harder to find them. When looking for comps, sorting by difficulty can also help.

If you're still struggling to see them, try beaching into the primary skill of the comp -- again, if you still can't see them, it's likely this primary skill just isn't high enough.