how do i unfreeze my pipes without making a mess?

Open the tap first, so the water has somewhere to go as it thaws. Then warm the pipe working back from the tap end towards the frozen bit - hairdryer, or a hot water bottle wrapped round it. No blowtorch and no boiling water poured on it. Before you start, find your stopcock. If the pipe has split you won't see it until the ice melts, and at that point you want the water off in seconds.