A wobbling ceiling fan is almost never dangerous, and almost always fixable in twenty minutes. The cause is nearly always one of four things, and they are worth checking in this order.
First, rule out the mounting
Turn the fan off and let it stop. Reach up and try to move the canopy, the cup that covers the ceiling bracket. If there is any play, the problem is there, not in the blades.
A ceiling fan must hang from an electrical box rated to carry a fan. A standard lighting box is not, and it is the single most common cause of a wobble that gets worse over months. A fan-rated box is braced against the joists and marked as such inside.
If you are not certain which one you have, this is the point to call an electrician. Everything else on this page you can do yourself.
Second, check the blade screws
With the fan off, work around each blade and check the screws that hold it to its bracket, then the screws that hold the bracket to the motor. They loosen over time with vibration, and one loose screw is enough to set up a wobble.
Tighten them firmly but do not force them. Over-tightening into a moulded blade strips the seat and makes the problem permanent.
Third, measure the blades
This sounds fussy and it is the step that solves most stubborn cases.
Hold a tape measure at the ceiling, next to the downrod, and measure down to the tip of one blade. Rotate the blades by hand and measure each one at the same point. They should all be within about an eighth of an inch.
If one sits lower than the rest, its bracket is slightly bent. Take it off and bend it gently back by hand. That is the whole repair.
Fourth, balance it
If the heights match and the wobble remains, one blade is heavier than the others. Fan balancing kits are inexpensive and contain a plastic clip and small adhesive weights.
Clip the weight to the trailing edge of one blade at its midpoint, run the fan, and see whether the wobble improves or worsens. Work around the blades until you find the one that improves it, then move the clip along that blade until the wobble is least. Stick a weight there and remove the clip.
It is tedious and it works.
What does not cause a wobble
A slight sway on a long downrod in a room with moving air is normal and not the same thing. Neither is a fan that wobbles only at top speed and runs smoothly at the settings you actually use; that is worth balancing but not worth worrying about.
Common questions
Is a wobbling fan going to fall?
Almost certainly not, if it is on a fan-rated box. Fans fall because of the box, not the wobble. But a wobble tells you to check the box, which is why that is step one.
Why did it start wobbling after years?
Screws work loose with vibration, and blades absorb moisture and dust unevenly. Both are slow processes, which is why the change seems to come from nowhere.
Can I just swap one blade?
Only for an identical one from the same maker. Blades are balanced as a set, and a single mismatched blade will always wobble, however tightly it is fitted.
Do heavier blades wobble less?
Not really. What matters is whether the set is matched to itself. Blades that come out of the same mould or are balanced as a set at the factory start true and stay true.
Choosing a fan that stays still
Two things help from the start. Blades balanced as a matched set, rather than assembled from a parts bin. And a motor that turns slowly enough that any residual imbalance never gets excited in the first place.
If you are replacing a fan that has wobbled for years, check the box before you buy, and consider a flush mount model, which has no downrod to amplify anything. Otherwise, the full range is here.