The luminous Lum can be used to brighten up dark spaces and nail those tricky vampires, but once it touches a surface it's stuck for good. They can also pass through glass without sticking to it.
The anti-gravity Lum is very useful and quite unpredictable. Think very carefully about how you latch onto a particular object. Picking it up just a fraction on the wrong side might result in an unlucky bounce when the object falls.
The transparent Lum can pass through glass. Not only that, it will turn even the most hard-wearing materials into glass, making it much easier for a cannonball Lum to clatter through scenery.
The stone Lum can be used to turn lightweight materials into very hard stone. This is very useful for blocking off laser beams that are in the way, but placed incorrectly you can end up being unable to complete the level.
The cannonball Lum is your bog-standard assault unit. These tough blighters can smash through even brick, although they'll travel further if you can fling them through lighter material, such as glass and wood.
Found an laser beam in your path!? Sometimes you're going to have to find a way through it so see if there are any nearby objects you can use to block the path of the beam, and make more progress that way.
Can't bust your way through a level!? Consider using one of your limited number of nukes. You can access these from the radiation icon in the top right-hand corner of the screen. Once activated, every vampire creature on the screen will evaporate, but you won't receive any star rewards.
Keep in mind that the order you can play the Lums actually gives you a big clue about how to solve each puzzle ie. if you're given a gravity-shifting Lum at the start, see if there's any scenery around that will open up a clear path ahead. A luminous Lum will almost always need to be placed well away from existing sources of light. Approach each level in this way, experiment a little, and you should have no trouble breaking down the challenge.