Some common ailments would be:

Sounds like the gearbox is shot, makes an awful grinding noise, especially when pulling away from rest.
Probably the rear engine/gearbox mount has gone. They don't come apart completely, just drop about an inch causing some metal to metal contact in the mount and posibly the exhaust to catch on the anti roll bar (if fitted). Parts are about 12quid from Fiat and take about an hour (including swear-break) to fit.

Clonking from the back end
Usually the rear suspension arm bushes gone, fortunately you can now buy them separately instead of having to splash out on a whole new rear arm from Fiat at over 100quid each. Try www.fastfiat.com for uprated Powerflex bushes at around 70quid for the full set of 4, or www.ggbengineering.com for some OEM type ones for around 8quid each. Bit tricky to do yourself unless you're a fairly competent spannerperson with a good range of inventive expletives.

Could also be worn out shock absorbers, or shock absorber bushes.

Also, some aftermarket exhausts, especially sports types don't fit (or get fitted) correctly, causing the silencer box to hit the rear suspension or shock absorber.

Clonking from the front end
Could be worn shock absorber top mounts or tired shock absorbers.
May also be caused by worn anti roll bar bushes (if fitted), worn out engine mounts, track rod ends or suspension lower baljoints.
If the noise occurs most at low speeds at near full lock, e.g. when parking, it could be badly worn driveshaft universal joints.

Screeching or howling when pressing the clutch
See Clutch topic.

Tapping noise from the top of the engine
Can occur in any engine if the oil is the wrong grade, too low or way overdue a change.

On 899cc engines - normally worn or clogged tappets. Cure involves removal and thorough cleaning, or more often replacement.

On 1108cc engines - not really very common but could occur at high mileages due to wear, especially if not well looked after. 1108's do not use the same system as the 899 engine, opting for bucket and shim type tappets instead.

Howling or grinding from the front or rear, alters with speed
Usually wheel bearings. Rear ones can last as little as 20,000 miles (especially how I drive;)) but are a simple DIY job to change and cost around 30quid a side for the bits. Front bearings are a bit more complicated to do.

Can also be caused by excessively worn brakes.

Squealing noise, especially when cold or damp
Usually worn or loose alternator drivebelt (fan belt)

On 899cc engines - hamster's wheel under bonnet needs a spot more grease ;):D

source: http://www.fiatforum.com/showthread.php?t=1858