What are the rules on naming a horse?
Unique Name: Each horse must have a unique name, which remains with them for life, to distinguish them from others in a race and facilitate tracing bloodlines and pedigrees.
Character Limit: Horse names cannot exceed 18 characters, excluding spaces. This means names like “Shutthefrontdoor” are acceptable, while longer names are not.
Protected Names: Around 3,000 names are protected, including winners of major races and famous retired horses. These names cannot be reused.
Name Availability: Before applying for a name, check its availability through the Racing Administration System or the BHA website. You can also download application forms for GB/IRE breds and foreign breds from the BHA website.
Registration: To be eligible to race under Rules or in point-to-points, a horse must be registered with a unique name. Most applications are processed and completed within a day, with over 13,000 names processed annually.
Basically anything within reason so long as it's within 18 characters. You can't buy a shitter and call it Frankel and you can't call them straight up expletives or similar but there's plenty semi rude ones knocking around.
My favourite 2 at the moment are Bolo Neighs and Mayo Neighs always make me chuckle when I see them run.