Although it is addressed here that the first and second movies are not connected in the way we're used to, there's some links that seem to be fairly weird between the three.
Why do the monsters, aliens, and plots of all three movies seem to be weirdly connected but do not share a consistent timeline or chain of events?
warning: SEVERE spoilers ahead
In the first installment of the saga, the city of New York's monster-prone neighborhood of Manhattan is (what a surprise) destroyed by a giant 6-legged arthropod-ish monster with big teeth, resistant to modern warfare and probably vulnerable to nuclear warheads.
In some weird manga called Cloverfield and released right before the first installment (but which only reached Europe later) the same monster lays waste to a Japanese city at some point before or after the incident.
In the second installment, we have a terror-esque bunker-not-so-safe trope movie about the world being laid waste by Russians, Chinese, and/or aliens.
One of the main characters escapes and finds out it was door number 3 the whole time, and things aren't going okay for our lovely earth, or at least, for the American east coast (what a surprise again).
In the third installment, we find ourselves on the verge of middle-of-21st-century energy crisis where the world is on the cusp of war for a couple of petrol drops, and a group of scientific international astronauts try to fix it by testing Hadron Collider-powered energy in space.
The experiment fails for 600-ish days, then it works, but it teleports the station somewhere else. Somehow a war starts on earth (by Russians, Chinese, or Germans this time), and the east coast is again laid waste. The husband of one of the scientists hides in a friend's bunker with a little girl rescued from a ruined hospital where we see a very weird shadow.
The scientists notice their earth is upside down, someone appears trapped in the electrical wiring of a station's panel, and weird phenomena starts attacking and killing everyone in a very gross way.
They finally realize they're in an alternate universe, and plot to get back to theirs by starting the machine again.
A lot of weird thriller-hide-n-seek moments later, our space heroes beam down to earth in a capsule, only to find the big brother of the Cloverfield (2007) monster doing stretches half-way-over the Philadelphia clouds.
I tried to come up with a decent explanation of what is going on in the Cloverfield "universes", but I would appreciate any official or non official side idea.