Here’s a screenshot of my use case:

Where the button is just a <button>Copy To Clipboard</button> element.
What’s the best way to accomplish this? I’m using jQuery.
I’m only concerned of having it work on modern browsers, and if it works on IE8> it’s an added bonus, but not 100% required.
Most cross-browser implementations use Flash to get past security restrictions. AFAIK, there isn’t a W3C standard for accessing the system pasteboard.
I’ve used Clippy in the past. It’s lightweight and fast and does just what it says on the box.