How about a wild swing with an axe that misses the intended target and embeds itself into the mast or into one of those figure of eight things that secure ropes? In my world I could imaging the axe man looking at his axe, seeing a length of rope whipping away up into the rigging and then looking up to see a huge block and tackle swinging down at him at head height and high speed.
The mast is likely to be thick enough that it would take several minutes of dedicated work to make it fail with a single axe.
...one of those figure of eight things that secure ropes?
Those are called belaying pins. They make really good impromptu clubs. Cutting ropes is actually a much better bet, although most of the running rigging (the stuff that gets pulled into place and belayed) will do little more than put the sails out of position, since getting the sails *into* proper position is what it's all generally used for. However, that can make a huge difference by itself in a sea battle. If the sails are just wind drag instead of actual control, boats and ships tend to turn up into the wind and stop, making them easy prey for any ship that *is* still under control. Granted, on a big ship with a lot of different sails you'll have to cut several ropes ("sheets," "braces" and/or "halliards") to lose very much control. You can achieve the same thing with a single cut if you can get below decks and cut the rope that connects the rudder to the wheel. In addition, a ship that isn't moving tends to wallow in the trough of the waves, making keeping your footing difficult.
All that said, let's talk about *standing* rigging (the ropes that "guy" the masts to the hull and brace them against wind pressure). In any kind of wind, a ship is going to have an immense amount of wind pressure on it, just because of the sheer amount of sail area. Therefore all the standing rigging on the windward side is going to tend to be under a lot of strain. So for example, if the ship is running before the wind (heading downwind) in a gale, cutting the mizzen stay (connecting the mizzenmast to the hull at aft center)
may dismast the entire ship all by itself. Sailing into the wind, you might get the same result by cutting the fore stay (connects the foremast to the bowsprit). Just to be clear, the stays run fore and aft, and their purpose is to keep the masts in the same position relative to one another and the hull. Cutting the shrouds (the things you see people climbing in pirate movies) will almost instantly make the ship uncontrollable as the masts and sails won't be able to keep their shape, it will all lean to the other side where the shrouds are still under tension. If you cut the shrouds on the windward side, the masts may all go overboard to leeward.
Also keep in mind that, because any line in use is probably under a lot of strain, if the normal run of the line is anywhere close to deck level (in other words it tends to run parallel to the deck rather than aloft), cutting it may send the free end of the rope scything along the deck like a huge blunt sword. A cable under strain can chop someone clean in half if they're in the wrong place when it fails. A cut line that runs aloft may snap upward and knock someone out of the rigging and over the side.... if they're lucky. If not, it knocks them out of the rigging and they fall to the deck. *Splat*
...seeing a length of rope whipping away up into the rigging and then looking up to see a huge block and tackle swinging down at him at head height and high speed.
That's quite possible, depending on how the block is rigged. Once the line that runs through it is cut and whips clear of the block, the block itself is likely to just be a weight on the end of a rope, at which point it acts just like any other pendulum.
Where on earth would I find a spanker boom?
At the foot of the spanker, of course. Duh.
(That was really helpful, huh?)
Note that a "boom" is attached at the bottom, or foot, of a sail, generally attached to a mast at one end, and adjusted into place by rope tension on the free end. That and they are usually fairly low and roughly parallel with the deck. Every day of every summer, all over the world, people in their sailboats get knocked overboard when the mainsail boom comes around and they aren't paying attention. Often it knocks them unconscious first, before sending them over the side. It's just possible that "wild booms" are the most common cause of drowning in sailing accidents. Cutting the ropes that hold the outer end of a boom steady turns it into an instant hazard to everyone in that area of deck.
Note that the wikipedia link that markc posted will lead you to sail plans and rigging plans, which may make it easier to visualize what goes where and what forces are at work on which parts.