Well, it is a bit of a punt out there, but you can help yourself an awful lot by doing a bit of homework first.
Once you have answered an advertisement or notice and identified an aircraft that holds an interest for you, find out what type of certification it has: Permit to Fly, Certificate of Airworthiness, Experimental or Standard Category. It could be a lot of hassle to get a C of A if the aircraft has been on any lower standard.
Check the engine hours run or left to run, and hours since the last top overhaul. Where was the engine overhauled and by whom? What mark of engine is it? There are lots of categories of Gipsy Major engines: basic 1, 1C, IF, 1H, 10/1 etc. Does the engine have a valve seat modification? Does the crankshaft carry mods 2495 and 2690? It could be expensive later if not. Does it have screened harnesses? Spark plugs are expensive. Are the heads to the latest modification standard and do they take 12mm or 14mm plugs? There are several permutations.
What sort of propeller is the engine swinging?
There are lots of possibilities: de Havilland, Hoffman, Evra, Invincible, Poncelet, Dunbar, Hercules, Granby or a range of lesser known types permitted under the appropriate certification approval. The type of propeller (pitch and diameter) matters a great deal: it is your gearbox! If you fit the wrong one it will limit your performance. You really want the pitch to be about 5ft 8in; below this the cruise is too slow. The Gipsy Major 1C and 10 Series can handle a bit more pitch due to their increased horsepower. Too coarse a pitch and the aircraft will be relying on the curvature of the earth to take off; too fine and the guts will be revved out of the engine whilst going nowhere. If in doubt, call somebody who knows. Consult the list of approved propellers published in the appropriate National Airworthiness documents. (For British owners, CAA Notice No 4).
Now to the airframe
Lesson number 1. Do not worry about the colour! However, look for a good, tidy, clean, cared-for finish.
Does the aeroplane sit level? Are the tyres pumped up? Is there a puddle of oil underneath her? How is the aeroplane presented: clean, smart and loved? Has she been used fairly regularly? Is all the fabric taught? Are there any wrinkles, especially just above the undercarriage attachment points? Are the cowlings in reasonable shape without patches on the patches? Are the instruments legible? Are the compasses serviceable and without bubbles? Is the floor clean and not oil-soaked? Is there a good wireless to the latest requirement with an intercom that works? Try it! Is the wireless part of the deal? Are helmets or headsets included?
Take out the back seat (4 x 2BA screws) and with the aid of a torch look in the rear fuselage for oil and muck, and especially corrosion on the tube work. Are there brown marks where the fabric wraps around the lower longerons? Can you see light coming through the fabric? That is a bad sign!
Look for play in the control column, fore and aft (worn bearings) and side to side (worn attachment pins). Check the throttle for back-lash: get a general feeling for what you have around you.
Look at the fuel tank carefully: check for leaks, repairs, crash damage, etc. If there are lots of patches, the tank will leak again. Does the fuel on-off cock leak? Look for stains on the fuselage port side. Ask to see the aircraft with a tank full of fuel: they do not always leak when half empty.
Check the turtle decking for dents and depressions. Tidy turtle decks are like neat engine cowlings: they make or break a Tiger Moth.
Have a good look at the undercarriage: this needs two of you. One gets a cushion, goes onto his back and lifts a lower wing (at the tie-down ring on the front spar), sufficiently for the wheel on that side to have all the weight taken off it. The second person tries to roll the wheel fore and aft, establishing the degree of play-in the universal joint. Lifting the wheel reveals play in the compression strut: more than half an inch and there is a problem.
Check the outside face of the floor for oil soakage and the oil tank itself for leaks, usually found at a cracked seam or from a split back wall.
Do the wing-walks creak when trodden on? Are the wing trailing edges fairly straight? Place your hands on each side of each rib (wings and tail) and gentle push up and down to see whether the glue has let go.
What sort of fabric is it? Linen, cotton, synthetic? Check it everywhere. What sort of paint is it? Nitrate dope? Butyrate dope? What is the top coat? Cellulose (easy to repair), dope (easy to repair), two-pack poly (a nightmare). When you push your thumb into the fabric are there cracking noises (tired fabric), and when you ping your finger on the panel is it like a drum, full of life, or is it flat and dull?
Is the aeroplane kept in a dry hangar? In multi-occupancy hangars check for signs of hangar-rash.
The Next Step
These are a few of the more obvious things to look at. Then you should see how the aircraft flies. Get an independent pilot who knows Tiger Moths to fly the aircraft with the owner or, after a double check of the insurance, by himself.
If you are satisfied so far and feel you would like to buy the aircraft, why not call in an appropriately experienced engineer, one who knows fabric covered wood and steel tube aircraft structures? He can run through the finer points of Airworthiness Notices and Technical News Sheets; identify whether metric or AN hardware has been substituted, assess hinge wear, cable condition, modification state, repair status; a general vetting as well as can be expected without removing all the fabric.
Much like old cars, old aeroplane purchase carries many pitfalls. Many of these can be avoided with diligent checks.
Perhaps the greatest single piece of advice is: do not, under any circumstances, be in a hurry.