I had two pets available for regeneration, a female and a male. The female had three trait points, and I regenerated her first- I have her star runes and had her carry fairy rings. But when she had been regenerated, the carried marking wasn't there. The male had one trait point, and I chose for him to carry star runes (I even took a screenshot in case it happened again) but it shows as "none" on the profile page. So the carried markings I chose just didn't seem to register after I regenerated the pets.

The carried traits are there, you currently can not see them because your pet is not genetically tested. After your pet is tested, the trait will be viewable. :)

Would you like me to test your pets for you?