A former coworker makes liberal use of American Express disposable credit card numbers -- proxy credit card numbers that you can request to give away to less than trustworthy merchants.
Privacy (https://privacy.com/) offers a similar service for those that want something that works with more than American Express cards or other such offerings from other card issuers.
I use this all the time. It's great.
Did not know about this! Checked it out, got really excited... and then discovered it's US only :-(
My Citibank card has the same feature, although the number generation applet requires Flash.
On a banking website.
In mid-2017.
Email addresses have been working fine for me. What I would love though is a way to easily alias my phone number.