I did a Sydney - Hobart today in a QF 734 and coming into Hobart we started doing a set of banking turns, the engines were kept at a higher then normal level and it seemed like a long time circling before the pilot came over the speakers telling us of a 'flap problem' and they were 'running the check lists' for a different landing.
That just sounds like a holding pattern, but with a bit of flap or slat (or gear) out, so that the aircraft is in a higher drag configuration than normal holding.
In the end I noticed that we had a very slow approach with greater use of engine thrust to control decent (from what I'm used to hearing).
Thrust is used to control speed, not descent, so it may simply have been a response to some wind variations. Why did you consider it to be a slow approach? Flap/slat problems normally involve faster, not slower.
Landing was fine, although firetrucks were on standby on the side of the runway.
They may not have even been asked for by the crew. But, perhaps it's better to be safe than sorry.
What might you guess a 'flap problem' might of been? I couldn't see the flaps. Could of been a stuck flap?
Let's talk about generic Boeing (they tend to be very similar) and see where we end up. I doubt very much that it was a 'stuck' flap. In fact, if a flap were not extended, then it would give you a much higher approach speed.
An airliner wing normally has high lift devices at both the leading and trailing edges.
Trailing edge devices are flaps, and they may just increase the curvature of the wing, or they may also extend backwards, and increase the wing area. They will basically increase the amount of lift available at a given angle of attack and reduce the stall speed. At the very high settings used on most landings, they also increase drag appreciably, and so allow the engines to be spooled up slightly, so that their throttle response is improved.
Leading edge devices come in a couple of forms. Slats or flaps (I don't know which the 737 has). The flaps increase the curvature of the wing, and so give more lift, whilst the slats work quite differently...basically they create a slot behind the slat, that the air flows through, and that in turn tends to help the air stay 'energised' and so not break away from the wing. So, a flap will give more lift at a given angle of a attack, whilst a slat will allow a greater angle of attack (wikipedia has a pretty simple explanation).
What can go wrong? Well, firstly the devices are generally powered by a couple of different power sources. They may be hydraulic, pneumatic, or electric (or any combination). Loss of any particular form of power means you'll need to extend them using the alternate power. That may be automatic, or might require you to use some rarely used switches and checklists (and might take a little time, as well as, in the first instance, forcing a go around). It's relatively unlikely that you'll lose the power to them.
Symmetry must be maintained. Having a device go out (or in) on one side, but not the other would be much the same as an aileron input, and would give a roll command. To avoid this, the flaps might be physically connected, or they might be electrically interlinked, with the system automatically removing their power if they are detected to be even slightly asymmetric.
The devices are normally split up into sections (inboard and outboard flaps), and various slat panels. Partially this is for packaging, to make them fit the wing section, but it also provides another level of redundancy, with, for instance, the ability to have one section faulty, whilst most of the system continues to work normally.
In my experience, Boeing flaps/slats are themselves quite trouble free, with enough backup extension options that moving them is rarely an issue. The biggest issue with them is actually a protection system. The asymmetry system, if it trips, will remove power to the devices, stopping them (actually only the symmetric panels) wherever they happen to be. Whilst this is a great idea, it actually puts another failure level into the system, as a problem with the symmetry protection can itself stop parts of the system from operating....so the flaps themselves might be fine, but the protections system isn't. That's a pretty benign failure though (and rare in itself).