Vibrato bridges have been problem children since their inception back in the mid 20th century, when dinosaurs ruled the Earth. But, they've always been such great fun that people have been breeding their problems and idiosyncrasies out in succeeding generations, until today when perfectly functional units that play nice and don't fight are as common as houseflies.
I'm unique, just like everyone else!
But, musicians are largely creative types, and as such are often slaves to their emotions rather than those pesky ol' laws of physics that just make their brains hurt. So, they continue to demand that old "vintage" (read: obsolete and/or problematic) designs be made available because, well, just because. Then, they take powerful drugs to fend off the realization that they're really just definite-purpose proxies put here on Earth for others to exploit for financial gain. But, I digress...
On the list of problems various vibrato bridges present, arguably the two most prominent and common to all since day one have been stability and repeatability. Stability refers to their ability to maintain consistent tension on each string and across a set of strings, which allows an instrument to stay in tune with itself. Repeatability refers to the bridge's ability to return to the strings to the tensile point they were at before any force was applied to move them, regardless of whether or not the strings were in tune to begin with.
The first vibrato bridge to truly accomplish this non-trivial feat was the Floyd Rose design, which rapidly rose to prominence among discerning users and remains a mainstay to this day. Once set up and tuned, you can beat the snot out of your strings and the thing is almost magical in its ability to keep all strings at their individual tensions as well as across the set, and return to a starting tension very accurately. This was accomplished with two key features: clamping (locking) the strings tight at or very near both ends of their speaking length, and using a two-point knife-edge fulcrum.
The locking feature was not without difficulties, however. The most obvious problem was once everything was locked up, you couldn't easily tune the instrument anymore. So, the mechanics were modified in such a way as to add "fine" tuners to the bridge which allowed small changes in tension without unlocking anything. That allowed for natural string stretch due to wear, although you couldn't change gross tunings if you wanted to go from, for instance, an open E to a Dropped D. The other obvious problem was the clamps themselves. Now you needed to carry and keep track of a tool or two to change strings, which also slowed down that operation in real time. Plus, all those linkages and pivot points made the thing more costly to manufacture, and tended to slightly dampen string vibration, leading to frequent accusations of being a "tone sucker". That last is probably more of a "sour grapes" thing than anything else, since the amount of dampening is almost certainly less than what you'd experience with any of the previous designs. But, you know... haters gotta hate even if it's baseless <grin>
Floyd's two-point knife-edge fulcrum was a no-brainer. Any time you have a pivot point under stress, you want the least friction possible to reduce wear and increase accuracy and repeatability. Almost all previous vibrato bridge designs didn't seem to consider this at all, which leads some to question the mechanical aptitude of previous designers (yeah, I'm looking at you, Leo).
Fast forward to today, and we've gotten rid of the locking nut in favor of locking tuners, which allows us to get rid of the "fine" tuners at the bridge and eliminate that maintenance and manufacturing cost and potential failure point. We've got better nuts with very low friction either due to rollers or material, so we don't have any string hang-ups there. Finally, we have better saddles, so variations in grip at the bridge aren't an issue anymore, either. We've got two-point knife-edge fulcrums on anything worth using, so repeatability isn't an issue. So, the Floyd is reduced to historical significance only, although you can still buy and install them if you like.
The whole point of this diatribe is this: no matter how little you use a vibrato, if it's not a good design, you're going to suffer. So, do the right thing or install a fixed bridge.