As soon as a competition reaches 75% prestige, it will become a rosette competition immediately (as long as the EC owner hasn't chosen to disable their rosette) and the next competition that runs will award the horse in 1st place with a rosette. However, if the comp drops below 75% prestige, the rosette is removed immediately and won't be offered until it can get back over the 75% threshold.

You have the option of enabling/disabling your rosette by heading to Equestrian centre > Competitions > 'Manage my rosette' (button in top-left corner)

I believe that a competition offers a rosette every time it runs at overall 75% prestige or higher. You can see this level on the individual comp page at the top left, under the Information heading. This will slowly increase based on how often the comp runs, and the quality of competitors for each individual run.

Note that this level seems to go up slowly, and come down quickly!

Thank you for your help