There are uncountable methods in investigation but mostly; they will use the following general methods.
Deduction by Material Evidence
Investigating the material evidence found where the crime takes place
What happened?
A man was killed in a closed chamber. The door was locked from the inside so it was mysterious that how could the murderer accessed to the room and killed the person. Even more, the fish tank inside the room was broken with no explanation. The people and even the police would think it was mysterious, if not related to ghosts or supernaturalism.
Use of Deduction by Material Evidence
Hajime Kindaichi noticed the glass shattered on the floor. He looked carefully at the material evidences in the room and came up with the conclusion: The murderer dropped and broke his/her glasses and tried to covered it up by breaking the tank so that nobody would be able to find the pieces of the glasses.
Common Material Evidence
1. Fingerprint
2. Trace of blood
3. Anything not belonged to the victim but found in the scene
4. Sharp and hard objects
In famous detective story, the detective has to be sensitive to the scene of crime, which has been set by the criminal in order to hide their trace, so as to deduce the method they commit crimes and identify them.