Most things on a used vehicle can be noticed by a trained eye if you know what you're looking for. Abuse, neglect, cover-ups; they all leave a trail.
Check the paint around all the doors, look for paint overspray; that's evidence of replacement body-work due to an accident.

Check the heat shield around the turbocharger and on the fire wall inside the engine bay; is it discolored? Brown? Removed?
If so, the car you may be looking at has been hot-rodded all over town.

Check the oil, coolant, the undercarriage for scrapes + dents, any leaks, and anything and everything to get the upper hand on the sales person or owner.
Double check all components and you'll find out everything you'll want to know on the car.

Other things to look for:

Oil Sludging / Low Oil Problems: The biggest failure in terms of negligence has been the oil level, and oil change frequency on these cars. Since the oil system affect the Vacuum Pump, Timing Chain tensioner, turbocharge feed and return lines, and critical engine bearings, a low oil condition can make for serious problems down the road. Read the oil dipstick, see if the oil is low or is extremely thick / chunky; if so, maintenance has been neglected and you'll want to proceed with extreme caution if purchasing the vehicle. If sludging occurs, you could eventually have problems with: 1.) the Vacuum Pump seizing which will shear off the camshaft timing sprocket and grenade the engine. 2.) Timing Chain tensioner not applying proper tension on the timing chain, allowing the chain to skip teeth on the camshaft sprocket, therefore ruining the engine. 3.) Turbocharger bearings not be supplied with proper oil flow which will cook the bearings in the turbocharger, causing seizure.

Timing Chain Tensioner [1.6T - N14 / N18]: When you first fire up the vehicle and there's a significant metallic rattle on the passenger side of the engine, that's the sound of a loose timing chain. The parasitic "Cold-Start Rattle" you will read about refers to the lack of proper tension on the timing chain or the chain stretched out beyond proper specification. There's been reworking of the timing chain tensioner and a replacement is easy to come by, but it should be required. There's also a TSB regarding if the chain is out of spec, it can be replaced, sometimes even out of warranty. Check maintenance history and any open TSB's at a local MINI dealer.

Valvetrain Coking / Carbon Build-Up: Since the turbocharged engine use direct-injection (DI) versus fuel-port injection (FPI), the fuel is directly shot into the combustion chamber. Because that is so, there's no fuel washdown on the backside of the intake valves. Due to the Positive Crankcase Ventilation (PCV) system on the engine, oil vapor is pulled from the valve cover into the intake system and can condense and build on the back of the intake valves. Serious coking of the valves will lead to obvious power reduction as the valves aren't properly closing for the combustion (power) stroke of the engine. What can be done about it? On the preventative side, make sure you're driving the car to bring it to proper operating temperature; short trips produce more vapor condensation. Once up to running temp, don't be afraid the bring the RPMs up to the rev-limiter of the engine; the 1.6T's love higher engine speeds, as it'll burn off most of the oily residue on the valves and keep the engine 'clean'. You can also install an oil catch can which with a passenger side PCV block port, will pull oil vapor through a condensing catch can first before being reintroduced into the engine. And finally, if the coking is so bad performance is hindered, the MINI dealer can do a walnut-media blasting service that uses crushed walnut shells to blast off any baked on carbon build-up; expect that service to cost several hundred dollars.

Dual Mass Clutch Assembly: A company called Valeo supplies MINI with the clutch assembly. During the early years of the 2nd-Generation, the clutches were incorrectly manufactured: the flywheel mass binding rivets were poorly installed causing excessive movement and play in the flywheel. That movement accelerated wear on the clutch disc itself, leading to a slipping clutch and eventually failure. When driving, take note of the available take-up on the clutch pedal. Engagement should be around mid-stride and should be pretty fluid. There is documentation out there if the vehicle has been covered or be given OEM replacement through the dealer; see the vehicle service history for more details.

Check Engine Light (CEL): Make sure all of the emission control circuits are intact and there are no Diagnostic Trouble Codes (DTCs) stored in the OBD system. This includes the Mass Air Flow (MAF) sensor, the ignition coil, Secondary Air Injection system, and other sensors. Check to ensure no other fault codes in other modules, like ABS, SRS (airbag), etc.

ABS System: Test drive the vehicle and perform a Panic Stop to ensure that the ABS system is in proper condition. If not, expect to pay a fortune to replace the system... avoid those at all costs, unless you've got a buddy in the business.

All Electrical Accessories: Make sure to nit-pick all of the accessories; power windows (all switches), power locks, AC, cruise control... the works! There's been some issues in the early 2nd-Generation models and the kinks are still getting worked out.

VIN Service History: If you're not convinced of what previous maintenance has been completed on the vehicle, grab the entire VIN number off the dash, and bring it to a MINI dealership. Their database will give you any insight of service maintenance records, if it's been brought in any MINI dealer stateside. You could use that information against the selling party to negotiate price if there's something that's required for replacement / service.

As for putting down the cash: Play hardball with the price! Flash the cash; that always puts you in the offensive and gives good results. Offer to pay what you want for the car, not what the seller wants. If they say "no", walk away from the deal. There's other cars out there; and besides, they (and chances are good) they'll meet your reasonable price before letting you walk away.

Really the R55 / R56 / R57 / R58 / R59's are good; consumer reports tend to be negative because of faults due to lack of proper ownership.
Honestly, some owners have run into problems, but that was due to abuse and obvious neglect on the car.

With the 2nd-Generation here's what you're looking at...
Let's break this down to see both side of repairs:

1.6T - N14 / N18:
* timing chain tensioner - oil sludging will produce failure and destruction of vavletrain
* vacuum pump seizure - oil sludging will produce failure and destruction of valvetrain
* turbocharger - oil sludging will occur if neglected or abused
* oil level / oil consumption - MINI specs consuming 1 quart per 1000 miles is typical; always keep engine oil to proper level
* carbon build-up - due to "grandma style" driving habits; can use catch-can to reduce oil vapor into intake system
* clutch assembly - early model faulty manufacturing and/or poor previous manual driver

All relatively easy if you take care of the car.
And if they do fault out, and you're mechanically inclined, you'll have no problems doing the many of the repairs yourself.

Best of luck!

- Erik