Gullfaxi - tips for finding comps?

Has he got 20 first place wins?
Most rosette comps are for horses with over 20 wins

After that he's best in cutting, barrels, and cross country

You also generally need to find the comp on your own; no auto comp. Because you are only allowed one entry into non-divine comps per day. I've on several occasions forgotten to turn off auto comp and he ended up in some random one that he'd either already won the rosette for or paid higher than any.open rosette comp.

I have over 20 wins on my and he does cross county races - I never have a problem finding races,, he runs with all the other horses

Have you tried sorting by the highest difficulty? Sometimes I have to try the different sorts to find one.

On non VIP I usually sort by highest kitty and usually there's a rosette comp somewhere.

The problem for this particular Gullfaxi is that he only has 10 wins.

The answer is to forget about rosettes and concentrate on getting another 10 wins then the cross country rosettes will all show for him.

Mine is good at gallop races but doesn't win rosettes for gallop just cross country

It's the 20 wins indeed. There are hardly any rosette competitions for horses with less than 20 wins.
As soon as he has 20, rosettes will be showing. Sorting the competitions in different ways is a good tip.
Gullfaxi does not have to win the competition to get the rosette. Competing in a rosette competition is enough, even when he gets 5th he will still get the rosette.
(Which is why he is the best guy for the 'win a rosette' objective )