Fetch Quest: Most of the games revolved around carrying objects from one end of the map to the other.
Magicland Dizzy was a particular offender; the official helpline ("calls cost 36p per minute during off peak time") even recommended starting the game by fetching as many items as possible and dropping them in a central location for easy access; specifically in Prince Charming's throne room, which is more or less central.
In Spellbound Dizzy you need to be carrying rocks to access half the rooms. You have to have the right number for the room you want to access, you can't put them down or they smash... oh, but you have to put them down or you can't pick up anything else or exit the room. Roughly half the game is spent just going back and forth to get more rocks. ARG.
The Grand Finale of almost every single game usually involves a map-long Fetch Quest from one end to the other, usually as far as it's possible to go, and usually back again. (Magic Land, for example; go to Hades, get Wizardslayer Trident; traverse map to the top of the Ice Tower, use Trident, get Zaks' Ring; traverse map back to The Cracks Of Gehenna, throw Ring into lava. Then start hunting for the Last Lousy Diamond...)
Follow the Leader: Dizzy's success led to many similar arcade adventure games, a lot of them also published by Codemasters. Examples include Murray Mouse: Supercop, Spike in Transilvania (sic) and the Seymour series.
Gaiden Game: This series got some very odd spinoffs, including Fast Food (a sort of Pac-Man clone) and Dizzy Down The Rapids (a Toobin' clone.)
Game-Breaking Bug (or possibly a Good Bad Bug): Treasure Island Dizzy in particular would sometimes misload, giving you mysterious floating scenery, jumbled music and holes in the floor. You could still walk around and explore, but would inevitably end up either freezing the game or falling through one of the holes and tumbling through the sky forever.
Incredibly Lame Pun: Egg-related puns feature heavily in the games, to the point of eggstreme eggsasperation.
Interface Screw: You travel through the looking glass for a couple of screens in Magic Land Dizzy, and the left-right controls promptly switch.
Similarly, in Fantasy World Dizzy there's an upside down world. Of course, if you're using a PC emulator, you can just Ctrl-Alt-Down your screen to flip it...
Inventory Management Puzzle: Pervades throughout the series, but the strict FIFO system in Treasure Island Dizzy is particularly heinous. Especially as a bug in the system would occasionally make you drop two items at once... bye bye snorkel *drowns*
Last Lousy Point: Even if you solve all the puzzles, most of the games require you to find thirty gold coins, or magic diamonds, or cherries, or something to get the real ending. You're required to use the pick up command on everything in sight to find some of them, which could be hidden behind a completely innocuous roof shingle or patch of grass.
Done rather insidiously in a few games where the tokens you need are also what refill your life bar, so you probably won't be picking them up if you're at 100% health.
The Merch: At the peak of Dizzy's popularity, the game inlays offered Dizzy merchandise including t-shirts, baseball caps and clocks for sale by mail order.
Nintendo Hard: Spellbound Dizzy in particular, see that page for details.
One-Hit-Point Wonder: In the early games anything dangerous was automatically lethal to Dizzy. Later games gave him a life bar but it went down really fast.
The second game, Treasure Island Dizzy, was foul with this. You're a One Hit Point Wonder with One Life - anything dangerous is instant End Of Game. Including those wretched cage traps hidden off screen in the trees.
Parental Bonus: In Crystal Kingdom Dizzy, Dizzy's girlfriend Daisy complains that he "messed up" her bedroom. Then once you get in there, you find a whip...
Portable Hole: You can pick one up, but as soon as you pick it up - and store it in your bag - there's a hole in your bag, so everything falls out of it. Including the hole.
Shout-Out: Various, ranging from the works of J. Milton Hayes, to the rest of the Codemasters library (the first game has the laser gun from the Oliver Twins' previous game Ghost Hunters), to pretty much every Fairy Tale ever written.
Magic Land Dizzy alone has puzzles that reference Pac-Man, the Billy Goats Gruff, Alice in Wonderland, Lord of the Rings, Aladdin, the legend of King Arthur, the Pied Piper of Hamlin, the Frog Prince, the Wizard of Oz, Sleeping Beauty and probably at least one more that's been forgotten.
Spiritual Successor: Seymour Goes to Hollywood was initially conceived as a Dizzy game, until it was decided that Dizzy didn't belong in a real world setting. Several more Seymour games followed in the same formula as the Dizzy adventures.
Super Drowning Skills: Treasure Island Dizzy introduced a snorkel, which sort of helps. Unless you accidentally drop it underwater, of course.