I'm currently in the transaction facilitation industry, and I was looking to fill an order of a buyer of mine. One of the prerequisites of the transaction is to ask for a Signature Message proving the other party owns their BTC Coins. Now, I have been surfing through multiples forums of people arguing over this topic. Some say it is 100% safe, if you don't sign using the same K Nonce, do it offline or do it on exchanges, some say there are risks involved. I find myself in between trying to have a clear idea of this known process in the BTC ecosystem, it's just a really niche feature not really used a lot nowadays. It can be done on Blockchain.com, Ledger, Electrum and Bitcoin Core and some other exchanges.
Signing messages is completely safe as long as you are using a safe route. If you make use of an electrum, a Bitcoin core, or any other noncustodial wallet that you have total control over that has the option and is open source (other developers verifying the legitimacy of the wallet) it's safe. Just don't go and input your private key or phrase into any online places asking you for such information, all in the name of the person the person you want to sign the signature message.
Bitcoin core, electrum, and other wallets that you mentioned above are not the same thing as exchange. It's not possible for you to sign in a signature message using exchange because they don't give you a private key; you can only do that using your wallet's private key, as that shows that you are the true owner of the said wallet.
