Just yesterday, a radio DJ was doing a trivia question. The answer went something like this:
One third of American children are forced to play musical instruments by their parents.
Can you spot the mistake?
The misplaced modifier above is the prepositional phrase "by their parents" which is presumably meant to add context to the verb "forced" since it is the parents doing the forcing. The way it comes across is that we don't know exactly who is doing the forcing, but it appears that they are being required to stand right next to mom or dad while practicing or performing. They're probably not being required to play their instruments right next to their parents but rather are being forced *by their parents* to play musical instruments. At least anyone who has had a budding band or orchestra student hopes as much.
Another good example is the following:
The doctors repaired the boy's leg that was broken during surgery.
So was the leg broken while in surgery and then later it was repaired? Or was it repaired during surgery after being broken elsewhere? If instead it was stated that during surgery the doctors repaired the boy's broken leg, that would clarify that it was repaired during surgery as opposed to being broken during surgery.
The biggest deal with this type of grammar error is that it doesn't seem like it's an error. In fact, from a syntax perspective, there's nothing wrong with it. It's the logical perspective that turns this type of unclear writing into a grammar issue.