Investigative method follows from what kind of act it was, because different acts leave different traces.
Crimes against the person — assault, homicide, sexual offences. The evidence is largely physical and medical, it degrades fast, and the first hours matter more than in any other category. Scene, body, medical examination, witnesses.
Crimes against property — burglary, theft, criminal damage. Traces are at the point of entry and on what was handled. Volume is the problem rather than complexity: these are worked with forensic triage and with intelligence about series rather than one scene at a time.
Financial and economic crime — fraud, money laundering, corruption. The evidence is documentary and lives in registers, accounts and correspondence. It does not degrade, but it is deliberately obscured, and the work is reconstruction rather than collection.
Organised crime — trafficking of people, drugs or firearms. The offence is a structure rather than an incident, and it is investigated as a network: who moves what, who pays whom, which crossing, which company.
Cyber-dependent crime — intrusion, ransomware. Evidence is volatile in a way physical evidence is not: it changes when it is looked at, and it can be destroyed remotely while the investigation is running.
What is the same in all of them. An account of what happened, tested against material that exists independently of that account. Every category above is a different answer to one question: where is the material that does not depend on somebody's word?
Source. UNODC, Handbook on police accountability, oversight and integrity and Crime scene and physical evidence awareness; Europol, Serious and Organised Crime Threat Assessment, https://www.europol.europa.eu/