Old, sun baked, dried up wiper blades
Usually when the wipers are dry and have no water (rain) to lubricate the blades. It also could be that the rubber blade is folded over. Squeaking, vibrating and bouncing is commonly called "chatter". It can be simply a byproduct of dry wipers on dry glass (no water (rain) to lubricate the blades). Most commonly and most likely however it's a sign of worn wiper blades that need to be replaced. As rubber gets older, it stiffens up and loses its flexibility and thus chatter begins. The last possibility, however unlikely, your wiper arms may be slightly twisted meaning that the wiper blade is not perfectly 90 degrees to the glass so the wiper blade doesn't properly "flop over" from side to side as the wiper arm sweeps. There are ways to adjust this, but I'd recommend starting with new wiper blades first and then looking into adjusting the arms if the chatter still persists.
Old or cheap blades.
The intermittent wiper relay is helpful when only a light rain or mist is present. It causes the wiper motor to only operate once every few seconds.
Most likely, either the linkage to the wiper has become disconnected (or broken), or the wiper pivot is stripped out.
If they are going on by themselves, check the wiper switch or the connections on the wiper motor. More than likely it is a bad switch.
The first thing to consider when experiencing chatter is replacing the wiper blades. Rubber gets stiff over time and fights the glass. See sources and related links below for wiper blade information.
The wiper relay switch for the rear wiper on a Ford Fiesta will cause the wiper to stop at the exact location where it is shut off. A hole in a vacuum hose will also cause this to happen, but the wiper will also run intermittently if it were a vacuum hose problem.
The wiper is just the part of the windscreen wiper assembly - usually rubber, although other materials can be used - that comes into contact with the screen. The Frame holds the wiper blade, and the arm connects the frame to the wiper motor. In flat beam wiper blades like the Aerotwin, the frame and wiper blade are combined in a single unit.
The wiper motor doesn't turn off until the wiper arm gets to the "park" location. If the wiper motor never detects park, the wipers will stay running.
In many cars, there is only a single wiper motor. One wiper arm is attached directly to the wiper motor and the other through a linkage. If one is working, the motor is operating but the connection to the other may be broken.
A bad motor can cause the driver side wiper to be slower than the passenger side. If one side is slower than the other, the blades can get stuck on each other.
Check your fuses. The wiper fuse may be blown.