This is a difficult question as the symptoms for a bad motherboard can also be seen as symptoms for other ailments as well.
Usually the motherboard goes dead and nothing starts or boots. This could also be the sign of a dead battery (on-board) or failed power supply unit....or even a dead CPU or as simple as a dead CPU fan or a bad connection.
In either of the above it is basically a proccess of illimination to determine what ails the computer.
Through experience...the first thing I do is make sure if there is any movement of the CPU fan when first booting up or if any LED on-board lights turn on...any movement shows that there is at least a start of power through the board.
Next thing would be to replace the battery ($2us ) and check the power supply unit...(PSU checker $10us).
After that is removal of the motherboard for a thorough search for burned out spots and expanded filters.
In the case that the computer does show some life but then goes out...then it is a check of all the add-on hardware to see if that is the proble....simplest way is to just remove everything except 1 harddrive and 1 stick of memory then see if it still does the same.
Again it is proccess of illimination to exclude everything that is possible to cause the symptoms and then make the determination of a bad motherboard.