What are the major problems associated with using inductors?

An inductor is one of the fundimental building blocks of electronic circuits. It is so basic, there really are no "problems" with them. Their behavior is known and modeled mathematically. 99% of the time they act exactly as expected.

Problems with inductors are usually problems with the design of the device the inductor is in. Inductors are like capacitors, except where cpacitors resist changes in voltage inductors resist changes in current. This requres the designer to think in terms of current, which is difficult for many to do. If you have current flowing through an inductor and try to break it instantly (like with a switch) the voltage across the inductor will spike to several thousand volts (because it is trying to keep current flowing, as it should.) Without something to absorb that (Like the condenser in a ignition circuit) this will cause arcing in the switch. Another problem is that inductors with an alternating current flowing through it will induce current into nearby wiring/inductors due to inductive coupling. Board layout, inductor type, and sheilding can alleviate most of this, but there is no way to truly get rid of it. It can be minimized great deal, though.

So, most problems are only problems in that device design. The problem of the voltage spike when current is interrupted is required for switching powersupplies and car ignition circuits, to name a few. Without it the powersupplies in your TV, computer, cellphone charger, etc., would not work. Inductive coupling is how Transformers work. Without it we wouldn't have, well, almost everything electronic we have today. It's not a problem with the inductor, it's a problem in the design.