Once For Each Revolution of the Wheels

Front or rear wheels:

The most common cause is dragging brake pads. The wheel may need to be centered between the brake pads or the brake may need adjustment. A problem that's often overlooked is a wheel that's positioned sideways in the frame, causing the tire to rub on a chainstay. 

Check tire seating. If suspected, deflate and reinflate. Better yet, try another tire. 

Check spoke intersections. The spokes can rub and create creaking and clicking noises. Squeeze pair of spokes around the wheel and see if you can duplicate the noise. A drop of light oil at each intersection will generally stop the sound. Of course if you have loose or broken spokes, you can expect noises from that.

You may have a hub problem. Remove the wheel from the bike, and spin the axle by hand. It should be smooth and not loose. If it is, a hub problem is unlikely. If not, you probably should overhaul that hub.

Rear wheels:Does it happen in every gear? If so, then it is probably one of the problems above.

If not: Look for foreign matter caught between sprockets - a stick or bunched up leaves, fishing line, etc. Look for a damaged sprocket tooth. You may have cassette or freewheel problems.

Once For Each Revolution of the Pedals

One of the most common causes is a loose crank arm. If your system is cotterless, as most are these days, you'll find a large nut or bolt in each crank arm that holds it to the axle. This should be kept tight. In fact, it should be checked for tightness every couple of months, or more often if you ride in cold weather. If a crank arm has come loose, even a little bit, there is a very good chance that it's inner surface has been distorted. If so, you may have to tighten the crank bolt often, yet the crank will continue to come loose. It will have to be replaced. If you really let it go, the axle will become misshapen also, and also need to be replaced. If you have let a crank arm become loose, you can try removing it with a cotterless crank extractor, clean and grease the surfaces of the axle and crank, reassemble it, tighten the bolt, and recheck it every couple of days until you see whether it is stable or will need to be replaced.

Note that some mechanics will tell you NOT to use grease between the crank and axle interface. That's ridiculous! Their theory is that the grease makes the crank slip too far on the axle. I've never seen that happen. But, without grease, your crank and axle can become stuck due to electrolytic action, and you can actually invite the loosening and distortion we've been talking about.

If you have an older bike with cottered cranks, you may find that the cranks become so loose that you can feel movement. The cotters should be replaced. If the distortion of the cotters is slight, you can file them flat and reuse them. 

A bike with a one-piece crank won't have this problem, so if you have noise that's not pedals or chainwheel, the problem is going to be in the bottom bracket bearings. These one-piece cranks often have bearing problems because the bearings are not well sealed against weather and dirt. 

On any kind of crankset, if you have screeching noises, it will be dry or damaged bottom bracket bearings most of the time.  

This generally means there is a problem with the pedals, bottom bracket or chainwheel, but don't rule out handlebar/stem or seat/frame issues since you pull on the handlebars and exert pressure on the seat that corresponds to revolutions of the pedals. You might remove the stem, grease the handlebar where it fits in the stem, grease the stem where it fits the fork, and grease the nuts and bolts, and see if the problem goes away. Same with the seatpost. Seats with springs are particularly prone to noises. Sometimes a spray of light oil will fix the problem.

One of the best way to check pedals is to borrow a different pair and see what happens. Pedals can have problems in the bearings, plates and screws or rivets, or toe clips. The most common source of pedal noise is the plates and fasteners. If held together with screws, try tightening them. A spray of light oil may reduce the noise on riveted pedals.

Chainwheels can produce squeaks or clicks if the chainwheel bolts are loose. 

Once for Each Revolution of the Chain

The problem is in the chain - no doubt about it. Look the chain over carefully for a bent, stiff or damaged link. A stiff link can often be lubed, and worked free by hand or with a couple of pairs of pliers. Try bending the chain just a bit laterally, as well as the way the link is meant to move, to free up rust or dirt. If the link is bent, it should be replaced. You should also wonder why it was bent. If the bike was thrown carelessly in a car or if it fell over that may explain it. But if not, it may have suffered some sort of shifting problem that could bend other links - and that should be investigated. In the meantime, don't force the shifting if it is not working well. If the link has come partially disassembled, run, don't walk, to the nearest bike shop and get a new chain! Unless you know that specific link has been mistreated, you can suspect that the entire chain has a problem and should be replaced. The thing you want to consider is what happens when you're pedaling hard up a hill and the chain suddenly breaks. This can be a serious let-down.

You may have a new problem when you replace a chain however. The chain and sprockets often wear together. As the chain stretches, the trailing edges of the sprocket teeth wear accordingly. Then, when you put a new chain on, it skips because it doesn't match the wear on the sprockets. This is more common on bikes made 20 or more years ago, but still can be a problem that requires replacing sprockets as well as the chain. 

When You Hit Bumps

Generally this is a suspension problem. Try lubricating pivots and tightening bolts. Look for loose accessories, or ones in which metal parts slap together such as luggage carriers and fenders. Chainguards are frequent noise-makers. Generally repositioning or bending the chainguard will fix the problem. Sometimes, the chain is too loose, and pulling the rear wheel back a bit will fix the chainguard problem. This is similar to chain slap, a problem in which the chain crashes down against the chainstay. Chain slap can be caused by a sticky freewheel or cassette. That will require replacement or overhaul. Many rear derailleurs can be adjusted to add more tension to the chain. On many bikes, some chain slap is natural. A layer of tape or plastic glued to the chainstay will reduce the noise and protect the paint. 

Randomly

Quite often a drop of light oil applied to the right place will fix the problem. Lubricate brake pivots (keeping the oil away from the rims and brake pads), seatpost parts, spoke intersections and so on. 

On coaster-brake and internally-geared bikes, the chain may be too tight. 

One of the best ways to find random noises is to temporarily replace items. Go with your suspicions. For instance, if you think it is in the seat or seatpost, borrow another seat and post and try them out. If that doesn't fix it, and you think it may be pedals, try another pair of pedals next. And so on, until the problem goes away.

Seldom

You have the biggest challenge with noises that happen only seldom. It can be like the proverbial car problem in which when the owner takes it to the shop, it doesn't misbehave, so the mechanic doesn't know what to fix. When test riding to find a noise that occurs infrequently, try pedaling hard up hill in every gear. If the noise is an occasional clunk, the chain and rear sprockets may be worn out and need replacement. That's because the chain is slipping a link at a time over the rear sprockets. This will most often happen in the smallest rear sprocket only. Depending on your tolerance for annoyance, it can be ignored for a while. This problem is not dangerous, but will get worse in time until it happens on larger and larger sprockets, and more and more frequently. 

Try turning hard left and right when looking for an occasional noise. Sometimes the problem will be exaggerated by the chain hanging at an angle, the wheels bearing weight off-center, or the accessories flexing into an unusual position. 

There's good news in small, infrequent noises: They seldom indicate a serious problem that needs immediate attention. If a thorough investigation turns up nothing, you can continue to ride the bike, and wait to see if the problem slowly gets worse, goes away, or ceases to bother you.