What would be require in this fun ring 
- pony club show?
Oddest pair of mount
bareback rider [of course bareback]
Natural rider
I can't give you definite help with those ones as I have not heard of them.
But I might take a guess that oddest pair of mount might be a small pony and big pony/horse pair together. Like the old fashion pairs where you have matching ponies of colour and height.
Natural rider, might not be to most proper but look natural? LOL sorry!
I just wanted to add to my last answer 
I have cut and copied our fun ring schedule. Just to give you an idea of the wording for the classes. They will have a specific ring dedicated to the fun ring and you should be able to enter more than one class I would imagine.
Each of my girls will be going in at least 7 of these events at their first show. So you should be able to go in a few as well
1. Neatest and cleanest combination of horse/pony - rider under 12 years
2. Neatest and cleanest combination of horse/pony - rider 12 years and over
3. Rider 6 years and under may be led
4. Rider 7 years and under 10 years
5. Rider 10 years and under 14 years
6. Rider 14 years and under 18 years
7. Rider 18 years and over
8. Best child’s mount Pony 14h and under - To be judged on manners and performance under saddle
9. Best child’s mount Galloway 14h and ne (not exceeding) 15h
10. Best child’s mount Hack over 15h
11. Ridden Pony
12. Ridden Galloway
13. Ridden Hack
14. Ridden black/brown Horse/Pony
15. Ridden chestnut Horse/Pony
16. Ridden palomino/buckskin/coloured Horse/Pony
17. Ridden grey Horse/Pony
18. Parent rider to be led by child under 16 years
1. Fancy dress rider/handler under 10 years
2. Fancy dress rider/handler 10 years and over
Good luck and have heaps of fun. It will all help you for the next one
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