I struggle to think of the amount of times I've seen that disapproving look between two people over which way round a fence panel goes. I've seen neighbours offer it a gesture of goodwill, toss a coin for it, show sickening displays of "No you have it" about it. Unfortunately I've also seen lots of neighbours fall out over it too.
So, who gets the good side? First off, what is the good side? Most fencing is made from vertical pieces (palisades, pickets and feather edge for example) and horizontal rails to hold them together. What is commonly referred to as the good side of the fence is the side with the vertical slats. Not all fences have good sides, Hit and Miss fencing for example has a slat either side of the rail alternately and looks the same either way round. Most have a side that is typically considered 'better' than the other though. It is however entirely subjective and I've had several clients who wanted the rail side to grow things along.
The best place to start is the deeds to find out which boundaries and fences belong to the property. You can often find a 'T' marked on the title plans to denote whose responsibility the fence is. A 'T' marked on either side of the line is a 'party wall' and joint ownership. If it is your fence and you're paying for it though you are entitled to have whichever side of the fence you like.
There are a few exceptions to that though such as if it is a listed property or in a conservation area like the Dedham Vale AONB. These are usually something along the lines of replacing like for like to maintain an aesthetic. I would also highly recommend a boundary fence that is open to the public, such as a footpath, has the horizontal rails facing inwards. This prevents the rails being useed as a ladder for easy access over your fence.
At the end of the day, the best thing I can recommend is to talk to your neighbour beforehand. Managing expectations can resolve a whole lot of possible disputes before they had ever happened!