Vibrations may be good when listening to music, but vibrations are often indicative of an issue with a component on the boat.
This is not a comprehensive guide by any stretch, but it will, hopefully, help you identify the source of any unwanted vibration, starting with the simplest causes.
Does the engine vibrate when it is running with transmission in neutral?
If Yes:
Worn engine mounts.
Inspect each engine mount in turn. Is one shaking more than the others? Is there excessive rusting or rubber residue adjacent top one or more of the mounts?
Remedy: Adjust or replace engine mounts.
Engine does not idle properly.
A rough-running engine can lead to excessive vibration. Run diagnostic checks in accordance with engine manufacturer's recommendations.
Remedy: Service or repair the engine.
Does the vibration occur only when the engine is in gear?
If Yes:
Fouled propeller.
Inspect propeller for rope, fishing line, weeds, or other fouling.
Remedy: Remove fouling and clean the propeller.
Bent, damaged, or missing propeller blade.
Carry out a visual inspection of the propeller.
Remedy: Repair or replace the damaged propeller.
Propeller not corrected fitted properly onto shaft.
Remove prop and check key, keyway, and taper for damage.
Remedy: Have the propeller lapped to the shaft to fit the taper correctly. Lap prop taper to ensure it fits correctly. Install new key.
Does the vibration increase or decrease at certain speeds?
If Yes:
Propeller may be out of balance.
Check prop with a dial gauge to ensure it is in round. On a sailboat with folding props, make sure that the blades open and seat correctly.
Remedy: Have the propeller serviced by a reputable propeller shop.
Cutless bearing is worn.
Grasp the prop and also grab the shaft near the bearing and try to shake/move it up and down and side to side with a lot of your strength. If there's discernable movement between the shaft and the inside of the cutless bearing then the bearing will need replacing, as it's worn. There should be zero fore and aft movement, which would be indicative of either a loose prop, a loose prop shaft coupling, or wear in the transmission.
Remedy: Replace the cutless bearing.
Is it hard to rotate the propeller by hand with the transmission in neutral?
If Yes:
Shaft may be bent.
If you suspect a bent shaft, have it inspected by a specialist facility.
Remedy: Remove shaft and have it straightened or replaced.
Propeller shaft strut out of alignment.
Check for visible cracking and twisting. Check mounting bolts for movement. Check hull around mounting bolts for vibration when you're running.
Remedy: Replace or repair strut.
Engine alignment poor.
Disconnect shaft from transmission. If shaft then turns easily, alignment issue is likely.
Remedy: Correct alignment between shaft and transmission.
Engine mounts may be worn and sagging.
Inspect each engine mount in turn.
Remedy: Adjust or replace mounts.
Is one shaking more than the others?
Is there excessive rusting or rubber residue adjacent top one or more of the mounts?
Remedy: Check and replace engine mounts.
Is there a visible wobble to the shaft inside the boat when operating at speed?
If Yes:
Coupling may be misaligned.
Separate coupling from transmission and check alignment.
Remedy: Adjust and/or replace engine mounts.
Engine mount(s) worn or improperly adjusted.
Inspect each engine mount in turn. Is one shaking more than the others? Is there excessive rusting or rubber residue adjacent to one or more of the mounts?
Remedy: Service, adjust, or replace engine mounts.
Shaft could be bent.
Specialist equipment is required to properly check for bent shaft.
Remedy: Remove shaft and have it straightened or replaced.
Transmission output flange distorted.
Specialist equipment is required to properly check for runout or defective coupling.
Remedy: Replace output flange.
Drivesaver, if fitted, is worn or damaged.
Remove Drivesaver from between shaft and transmission coupling and visually inspect for tears and distortion.
Remedy: Replace Drivesaver.
Congratulations!
If you've made it this far, you should have been able to fix, or at least identify, the issue that was causing those unwanted vibrations. On the other hand, if you have not been able to isolate the issue, it may be a good time to seek expert help from a competent yard.
But at least you gave it a shot, and hopefully you learned something about your boat along the way.