event.clipboardData.setData in copy event

Clipboard APIs were indeed in active development as of 2016, but things have stabilized since then:

Using event.clipboardData.setData() is supported

Changing the clipboard with event.clipboardData.setData() inside a 'copy' event handler is allowed by the spec (as long as the event is not synthetic).

Note that you need to prevent the default action in the event handler to prevent your changes from being overwritten by the browser:

document.addEventListener('copy', function(e){
  e.clipboardData.setData('text/plain', 'foo');
  e.preventDefault(); // default behaviour is to copy any selected text
});

To trigger the copy event use execCommand

If you need to trigger the copy event (and not just handle the copy requests made by the user via the browser UI), you must use document.execCommand('copy'). It will only work in certain handlers, such as the click handler:

document.getElementById("copyBtn").onclick = function() {
  document.execCommand('copy');
}

Modern browsers support both methods

Firefox supports both clipboardData in the copy/cut/paste events (since Firefox 22) and execCommand('copy') from user actions (since Firefox 41)
Chrome also supports both (the latter was added in Chrome 43 – or perhaps 42?)
caniuse.com claims that Safari 12 has complete support, versions up to 9.1 didn’t support execCommand('copy').
MS Edge Platform Status lists IE/Edge as supporting the Clipboard APIs as of 2019, unlike when this answer was first written (2016).

https://github.com/garykac/clipboard/blob/master/clipboard.md has a compatibility table for execCommand(cut / copy / paste).

You can test this using the snippet below, please comment with the results.

More resources

Specification: Clipboard API and events
The Definitive Guide to Copying and Pasting in JavaScript (2014) – more information on clipboard API interoperability in browsers, including support for the “copy”https://stackoverflow.com/”paste” events without a selection and support for multiple formats.
Pages tagged “Clipboard API” on MDN

Testcase

window.onload = function() {
  document.addEventListener('copy', function(e){
    console.log("copy handler");
    if (document.getElementById("enableHandler").checked) {
      e.clipboardData.setData('text/plain', 'Current time is ' + new Date());
      e.preventDefault(); // default behaviour is to copy any selected text
    }
    // This is just to simplify testing:
    setTimeout(function() {
      var tb = document.getElementById("target");
      tb.value = "";
      tb.focus();
    }, 0);
  });
  document.getElementById("execCopy").onclick = function() {
    document.execCommand('copy'); // only works in click handler or other user-triggered thread
  }
  document.getElementById("synthEvt").onclick = function() {
    var e = new ClipboardEvent("copy", {dataType: "text/plain", data:"bar"});
    document.dispatchEvent(e);
  }
}

<html>
<input id="enableHandler" type="checkbox" checked>
<label for="enableHandler">Run clipboardData.setData('text/plain', ...) in the "copy" handler</label>
<p>Try selecting this text and triggering a copy using</p>
<ul>
    <li><button id="execCopy">document.execCommand('copy')</button> - should work.</li>
    <li><button id="synthEvt">document.dispatchEvent(clipboardEvent)</button> - should NOT work</li>
    <li>with keyboard shortcut - should work</li>
    <li>or from the context menu - should work</li>
</ul>
<p>If the "copy" handler was triggered, the focus will move to the textbox below automatically, so that you can try pasting from clipboard:</p>
<input type="text" id="target" size="80">

Async Clipboard API will provide a simpler way to manage the clipboard

When implemented, navigator.clipboard will let you write code like this:

navigator.clipboard.writeText('Text to be copied')
  .then(() => {
    console.log('Text copied to clipboard');
  })
  .catch(err => {
    // This can happen if the user denies clipboard permissions:
    console.error('Could not copy text: ', err);
  });

Chrome 66 starts shipping a partial implementation, and they’ve published an article about the new API.