If you are ON the road, you are considered a vehicle in pretty much any place, UK, US or AU. So you must follow the road rules as if you were a vehicle - and that includes riding on the side that you would drive a car on.
I have spent years riding from the country areas into suburbs, residential, and retail areas, talked to every level of government, and understand how confusing this can be. Here are some (potentially) interesting points from my journeys in this area of NC...
- Horses are considered livestock. In the NC laws, you're not a vehicle or a pedestrian, I.e. There are definitions of what a horse back rider IS NOT, but there is no definition of what a horse back rider IS.
- Outside the agriculture and equine liability laws, there are few mentions of horses at all. Most rules are based on 'common law' (i.e. Legal case history) and not statutes (written laws).
- Many of the equine related state laws are from 100 years ago when cars were new to the world and injuring or stealing a person's horse was depriving one of their transportation and/or method of earning a living (and hence a felony offense).
- The newer laws tend to be at the town/city level, especially in areas where development is pushing into the countryside. For example, many towns prohibit horse riders (like bicycles and skateboards) from the sidewalks and parks. Some towns now have horse rider helmet laws for minors.
- Law enforcement folks (police/sheriff) typically are unfamiliar with the laws. Dealing with horse/riders is just not something that comes up very often.
- In NC, there is no requirement to clean up if your horse 'soils' the road, even in residential areas (town/city laws may apply). This has come up a number of times in my area where the suburban type developments are pushing outward (we are not in a town/city, and I've made a number of 'city folk transplants' very upset).
The bottom line is that after researching your applicable laws, using 'common sense' (i.e. What a reasonable person would do) is your best bet, and being extremely cautious is a must.
I love to ride the roads and be able to ride from/to anywhere I choose, and I know many other people do too.
Be careful and safe, good luck, and enjoy your riding where ever that may take you.