There are many miles of unpaved (or extremely badly paved) road on the Military area (SPTA), and a lot of dirt and mud tracks. I've been riding/driving there for years and the only time I was ever stopped was on a byway that isn't often used, in my Jeep with Northern Ireland plates.
The land wardens are friendly and uderstand that it's very difficult to tell which of the unpaved roads you can or cannot use. I've often used ones that don't have a right of way and passed police or wardens, sometimes several times on the same day, who didn't say anything.
Mudplugging will probably only be tolerated on rights of way though, and using vehicles away from tracks is never tolerated.
The SPTA is split into four parts:
The eastern area between netheravon and tidworth which is not used for live firing is always open to the public except during very large manouveres and has long sections off 'mud' tracks and miles of unpaved roads. This area is good because there is a large hill at one side, and the perimiter track is quite high up so you can easily get your bearings. the downside is there can be a lot of vehicle manouvres so you sometimes have to stay out of the way of the military.
The central area between Netheravon and Tilshead is an artillery firing range, the unpaved roads here are potholed and more interesting and can be very remote when they cross the middle of the ranges, and there are numerous 'military curiosities' by the roadside (not to be interefered with though) and some very tough mud tracks. However much of this area can be closed during live firing, call 01980674 763 for a recorded message detailing safe periods, updated Friday mornings. The Bustard Inn is at the southern end of the major route across this area.
The Western most area is divided into northern and souther sections. The North-Western section, between Westbury/Warminster/Tilshead, known as 'Imber range' is always completely closed to the public, except for a few days a year when the paved roads are open to the public (usually Xmas and Summer bank holiday periods). Very pleasant countryside but no 'off roading' to be had in this section. Well worth a visit when open as the village of Imber is an interesting curiosity (See Welcome to Scott Wylie's Photography Emporium - powered by SmugMug
The South-Western section is similar to the Eastern section, but has the Copehill Down facility in the middle, which is an urban warfare training site comprising several streets of houses and a fake middle-eastern village amongst other things. I don't think there is much off-road around this section but Copehill Down is an interesting curiosity (See Picasa Web Albums - Scotbot - Copehill Down
). Be careful around here as there can be a lot of troop and vehicle manouvres, a parachute drop zone, and you may cause a nuisance to training exercises.
Hope this helps...