It can be useful to prove someone controls a certain amount of bitcoin balance, using the sign message feature.
But it is also useful to consider a fresh transaction, that intentionally "burns" a specific amount of money by giving it to the miner as a transaction fee. This helps support a public service -- bitcoin transaction validation -- while assuring that real cost, real effort was spent to create an identity.
The keys used to spent the bitcoins in question provide another sign-message entry point.
That is an intriguing idea, since you can't know which miner will get it. I think people would prefer, though, to retain possession of the funds, esp if it's a large amount. That way, if you decide to stop using a particular identity, you haven't lost the money associated with it.
Of course, if you are a miner or pool operator yourself, you can just put the transaction into your own block instead of broadcasting it to the network. So it doesn't cost you anything.