Spoilers redacted. Select to view.
The following seem to form the most common sorts of problems you need to solve, and I've described the viable (and some not so viable) approaches to dealing with them that I've found. I wonder if I'm missing some major stuff, because I find that I keep falling back to a few similar items over and over again. If there are five different items to solve a puzzle that all work identically, do they really count as five solutions or as just one? Am I suffering a lack of imagination?
Crossing gaps: PEGASUS, SPACESHIP or occasionally HELICOPTER. WINGS or JETPACK are not good because you can't hover. It seems it's almost never a good idea to build a bridge, because it requires using lots of words to build and stick together all the pieces, it'll be fragile anyway and sometimes it seems like stuff randomly falls through my bridges made of PLANKs and GIRDERs anyway.
Killing stuff: ANTIMATTER, BLACK HOLE and INFINITY are all the same and kill everything near to them. MINE, TNT and DEPTH CHARGE are a bit more precise. LIGHTNING is powerful. GOD, SATAN, DEATH, CTHULHU etc. can kill most things on your behalf.
Digging: SPADE. (Use with PEGASUS to dig into high walls and ceilings.) I can't find anything more effective. It doesn't seem to recognize "pneumatic drill", and a DIGGER is no faster than a SPADE but extremely bulky. You can also use explosives to dig, but you'll generally need a lot of them, whereas one SPADE will last a whole level.
Doing anything underwater: FLIPPERS, SCUBA, WETSUIT and SNORKEL all do the same thing. (The only difference is that the scuba tank floats and the rest sink when you're not wearing them. Um, is that the right way round?) I wouldn't bother with the SUBMARINE, it's too bulky to be useful as far as I can tell. In theory you can fetch items from underwater with a FISHING ROD, but I've never found a level where that was more economical than the other more general purpose items.
Moving large inanimate objects: Tie a piece of ROPE, HAND CUFFS, NET, WIRE, etc. to it and drag the other end, possibly with a vehicle. It's slow, but I still haven't found a better solution. I wonder if I'm missing something here, because I've found levels involving moving crates to be really dull and tedious.
Moving people: Put them on a PEGASUS or in some vehicle. Tow it like an inanimate object. Again, maybe I'm missing something?
Moving animals: Lure them with food or something they like. I haven't found many clever things here. A COW will follow a BULL. If all else fails drag them with ROPE.
Stopping creatures from killing stuff without hurting them: This seems to form a large part of the game, and I generally can't spot clever solutions. (Exception: I stopped an angry WITCH by summoning a WARLOCK to be her friend.) Most of the time I just resort to dragging the offensive creature to a spot where it cannot do too much damage. This isn't so fun.