I can't figure out how to actually trigger it to start measurement in single-shot mode (nor how to reset it if a measurement is present).
Don't touch the "auto" button.
First, set up the triggering to trigger on the rising edge of your pulse. This may be as simple as setting the trigger level about halfway between the low level and high level of the signal, and leaving the rest of the triggering set at default (usually, default is a rising edge trigger).
When triggering is set up, push the "single" button on the front of the scope. The scope will sit there and wait for a triggering event. When it sees one, it will record what it sees and display it on the screen, and keep it frozen there until you push some other button on the scope.
Once you have the pulse on the screen, you may adjust the time base or the vertical scale to zoom in or out or pan around on the waveform that you sampled. If you change things too much, you might notice you start to lose resolution, or you run off the end of what was sampled. This happens because you're zooming around in what was previously sampled, and it was sampled at a fixed rate and vertical resolution. If you want to resample, just hit the "single" button again, and then trigger another pulse on your DUT. This will measure the pulse again, at whatever resolution your scope is now set to.
Play with this until you're really comfortable with it! It's one of the most powerful features of a DSO, when compared to an analog scope. It can be useful for things like capturing devices talking to each other via RS-232 or similar. Also, for capturing start-up glitches on power supplies, or any other single-shot event.