Digging: holes big enough to bury someone in and holes beside fences seeming too small for your dog to be able to wriggle through, but he does. There is a simple way to stop (usually) the digging.
When you come upon a new hole, fill it with that days' fresh doggie droppings that you usually toss away. They will soon dry out but will still be effective. Leave it uncovered a few days before filling it with dirt. For fence line diggers, line the bottom of the fence with the droppings. For areas where the dog can gain access to the street or for determined diggers, you will need to install an added barrier. The cheapest and easiest can consist of 2 foot wide one or two inch mesh chicken wire. Staple or tie one edge of the wire to the fence at least six inches up. Then bend it so it follows the fence to the ground then lays on top of the ground at least a foot out. You may want to anchor this into the ground with stakes or dirt or grow a ground cover on it.
When a dog has jumped a fence, he won't forget it, so you must take steps to stop it immediately before someone or some dog gets hurt. Other than tearing down a low fence and replacing it with a higher one, you can extend the original height with chicken wire. Tie six or eight foot tall posts to the original posts and attach two foot wide chicken wire to the upper section all around. Also anchor the bottom of the wire to the fence in case the dog tries to go between the fencing. If you can't stand the sight of it, it can be effectively concealed with bamboo or reed fencing or a growth of vines.
For both digging and jumping, you can try installing electric fence lines across the bottom or top of the fence or about 6 feet in from the fence. Do not use a cattle charger, but an alternating charger which sends pulses of current, not a constant stream of current. This is essential. Dogs have a higher salt content and when they come in contact with electricity, they cannot turn loose (in the case of biting the wire, I personally viewed this with my own dog). The pulsing breaks the electric current and allows the dog to free itself. Some dogs will learn what is and is not hot and will clear (jump) the wire. If that happens, a higher or double wire is needed. Or a higher fence, possibly with an overhanging stretch of wire angled into the dogs' yard.
A word about invisible fences...they can deter your dog from leaving (or returning) to the area but it will not keep out stray dogs that can enter and mate with or fight with your dog, nor will it keep out strangers. And they are effective only when the collar is worn.
