It's widely accepted across Equestria that all three races of ponies are of equal value and merit, and almost everypony is accepted everywhere. If you're barred from entry into something, there's probably a good reason, aside from the nature of your birth.
There are two exceptions to this rule, where a citizen can be cast aside based on their race. Both of these are places where only one race is allowed, not due to hatred or dislike of the other races, but just because the requirements bar everypony else from entry.
The first is Celestia's School for Gifted Unicorns. The school is for unicorns looking to improve their magical capabilities, and as such, the entrance exam requires that the applicant already have at least some amount of talent with magic. The other is the Wonderbolts' academy. Pegasi come here to test their flight skills and learn to become better fliers.
The paperwork for these places does not mention anything about the race of the applicant. A pegasus can absolutely fill out all the forms required to attend CSGU. But without a horn, they would certainly have a hard time casting spells, and probably couldn't make it through the entrance exam.
And it is well-known, as well, that earth ponies cannot fly.