Most of us are more worried about waistline expansion than that of our water. Have I said before that home inspection is a lot like medicine? You are going to have different valid opinions, much is based on the condition of the particular patient and accepted practice will change over time, especially with new procedures. There are some hard facts that won't change: in this instance, however. 1) Water expands when heated; and, 2) Water is heavy.
These expansion tanks have bladders in them that help relieve undo pressure in your pipes. (They also help absorb shocks from water hammer, which is sudden bursts of water shaking the pipes.) They have been a code requirement since 2006: Sec P2903.4.2... which says that some sort of device for dealing with water expansion is necessary if something is preventing pushing back on the public supply, like a pressure regulating valve:usually found above the water main) . They will always have some water and some air in them until the bladder fails when they will fill with water. The installation instructions consistently say that a vertical installation on solid pipe is OK. Some say this and then show pictures of horizontal set ups. Some practitioners recommend upside down installation so the bladder stays wet and then lasts longer. Some directions say install them away from the water heater, i.e., not directly above it, so if it leaks it won't damage the water heater.
I recommend them on old installations when the fittings above them are leaking and suggest they never be held only by piping because a five gallon tank full of water is like a toddler dangling on your plumbing. It just makes sense to take a little extra effort to prevent much bigger potential problems later.
Tellingly, M.2003.1 in the 2012 VA code requires expansion tanks on boilers: "supported to carry twice the weight of the tank filled with water". So it seems obvious that expansion tanks on water heaters should be as well supported.
The installation below seems like a real smart way to do it!
Above is an 11 month old expansion tank with failed bladder (full of water) stressing the joint where it connects to the water heater. The lateral copper pipe is connected to CPVC, which allows the tank to sway back and forth. Should the connection breach, nothing would stop the flow of water.
Here is a recent installation that uses a steel bracket attached to a stud, with a tight steel strap around it. very nice!