I figure I'll go Jeopardy style and give out some answers to common problems before I get the questions. Some of these can probably go in other forums, but what the heck! I'll update it frequently, so PM me if you have additions.
General Tips
+Check your cables
-Check to see that cables and cards are plugged in properly.
-Always try unplugging and plugging it back in, to ensure proper seating.
-Look for dust contamination. When unplugging a card, inspect for dust that can lodge in slots [frequent with PCI/AGP cards].
+Check your jumpers
-Sometimes when swapping devices or doing major overhauls, jumpers can be neglected, either from being temporarily removed or from not being repositioned correctly [e.g. master to slave].
-Some motherboards have jumper settings that you haven't encountered before [such as +5v USB or PS/2 standbys].
+Check your manual
-Yes, it's always worth a shot.
+Check for grounding problems
+Motherboards
-Some solder jobs leave longer leads than others. Take some electrical tape and put it a) Between the standoffs of your case and your motherboard and/or b) on the backplate of your case [NOT on back of the motherboard, as it can insulate heat].
+Boards and Cards
-Make sure no wire leads are touching your circuitboard [this goes to you modders that forget to connect or insulate your wire leads].
-Make sure they are seated correctly. This is especially true of RAM and PCI cards, where one side can be slightly off, either not contacting at all or contacting the wrong contacts on the board, leading to shorts.
+Drives
-Hard drives in particular, which aren't electrically insulated on the bottom, can occasionally short against the bottom of the case, bottom of the drive cage, or top of another drive. This also holds true for rheostats, system monitors, and fan controllers.
Floppy Drives-The most common issue is the little metal slider snapping off. Using a standard 12-piece toolkit, use the small screwdriver to prop the door open. Depending on ambient light, use a flashlight to find the offender. Use tweezers to gently remove the metal scrap from the drive.
Optical Drives
-Most problems stem from dust contamination. Don't use compressed air, as it can push the dust further INTO the drive. Try to get a mini vacuum cleaner, it will come in handy with the rest of your system as well.
More to come on other hardware.
