The BEST fence is probably going to be a 6' solid wood privacy fence. Why? Well, 6' because it is then high enough that most dogs can't/wont jump it and few people can look over the top and see the dogs (thieves, etc). Privacy fencing is good because the dogs can't catch their feet, collars, etc on anything and they can't see things to bark at or stress about (dogs walking past, wild animals, the neighbor kids playing, etc). That is going to be an expensive fence, however.
Probably the most bang for your buck is going to be either 5-6' chain link (with privacy slats if that is a concern) or the really cheaper but labor intensive and not so pretty livestock panel fencing. If you go the later route, PLEASE ensure that you are getting panels designed for goats and smaller stock so that the holes are too small for the dogs to escape or get hung up trying to escape.
As for permitting, that depends on city/local laws and any relevant HOA or other policies.