1 Loose/ineffective brakes:
How to detect: You'll need to apply more pressure on the handle before you start to slow down, due to a loose cable, or pads that are either worn down or not aligned to the rim of the wheel. It might sound easy to detect, but remember to check frequently. What to do: Most problems can be offset -- temporarily, at least -- by adjusting the hollow bolt. That's located at the front of your bike, where the cable emerges from the brake levers.
Dangers of ignoring: Aside from the obvious risks, a poorly aligned pad can also wear away the side of your tyre.
2 Unresponsive/ jammed gears
How to detect: When you shift gears, nothing happens.
What to do: This is one for the experts. A gear problem can come from a rusty cable, a worn chain and cassette or a worn shifter -- it can be hard to detect what is what.
Dangers of ignoring: The gear system is tightly integrated, so damage to one part can easily spread. Also, a limited selection of gears can make cycling quite tricky.
3 Loose/bent spokes
How to detect: You'll probably hear a rattling sound from the wheel first -- then, on inspection, you'll see that one or more of the spokes have bent and possibly come unstuck. You'll also probably notice that the wheel is touching the brake pads as it rotates -- that means it's buckled.
What to do: This is another one for the experts. Messing around with spokes can seriously damage the wheel if done incorrectly, and end up costing you a lot more in the long run.
Dangers of ignoring: Spokes support the structure of the wheel, so riding without them can do some serious damage.
4 Damaged tube
How to detect: Your tyre is either completely flat, or is going flat soon after being pumped up.
What to do: You can prevent damage to the tube by keeping it properly inflated, and making sure your tyres don't run too thin. A repair kit will do the job for small holes, but anything more serious will require a new tube.
Dangers of ignoring: Not only will you be very uncomfortable riding on a bike with a flat tube -- you'll also destroy your wheel quickly.