Twisters: on more than one occasion I've built my first city and discovered that Twisters spawned regularly (like, every 15 - 25 turns) right next door. It's more than annoying when your units need Ruby Slippers to get more than 5 tiles away from your Capital without getting a house dropped on them.
Volcanoes. I don't put anything next to a volcano except the magically-exempt-from-disasters Wonders. If there is a resource next to a volcano, I ignore it as putting an Improvement there is asking the Volcano Gods to smack it a few turns later - and keep smacking it Forever.
On the other hand, just last night in a game, for the first time I got a note that one volcano had become 'dormant' and there was no more danger of it erupting - first time I have ever seen any disaster 'turned off' by the game as opposed to having to alleviate the effects by my own (or the AI) actions.
Floodplains. On Level 2 I have seen the same river flood 4 times in the first 20 turns of the game. Luckily, it wasn't my city that was settled on its Floodplains! I am grudgingly wiling to put Improvements on Floodplains, but not Districts, and if I have more than a few Improvements on the Risky River Bottom, I try to keep a Builder handy to repair the inevitable Flood damage.
Hurricanes. As far as I'm concerned, they are the Game's way of telling you that Coastal City Sites are not your friends, and that Poseidon is a thorough-going SOB.
Storms, as in Dust or Blizzard. Have actually not had that much trouble from them. When I play on a lot of blizzard-prone Tundra, I'm usually playing Russia, so I have my Magic Petrine Anti-Blizzard Cream to at least protect my units. In the desert, I admit that I assume anybody crazy enough to go there deserves what they get (having lived in a few deserts in my time)
Drought is a bit puzzling. It appears to tear up Improvements on a random basis, but I've never seen a unit affected. Are all my Recon, Ranged, Melee, Cavalry, and Siege Units accompanied by trains of water-bearing Camels? It appears so . . .