Can any desktop browsers detect when the computer resumes from sleep?

I don’t know of any direct method to do this, but one way you could get a good idea of when it happens is to set up a setInterval task that runs, say every 2 seconds, and stores the time it last ran. Then check to see if the last time it ran is very much older than 2 seconds.

var lastTime = (new Date()).getTime();

setInterval(function() {
  var currentTime = (new Date()).getTime();
  if (currentTime > (lastTime + 2000*2)) {  // ignore small delays
    // Probably just woke up!
  }
  lastTime = currentTime;
}, 2000);