This is a broad category with tons of flexibility. Depending upon what character role you are assigned, you'll have a big range of choices for any of them, which should be fun for you.  The thing that holds this group together is that your characters are often in the field, not wanting to be noticed, and ready for conflict with their target in some cases (bounty hunter, spy, assassin). 

For detectives, you can go simple business casual with a fedora or deerstalker cap, or just go all in with a Sherlock costume. Another alternative is the tactical police costumes. 

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Bounty hunter roles have a huge range, as well - you can do a Dog the Bounty Hunter-inspired biker costume with bandana's and biker boots, or opt for more of a spy trench coat theme.  Another option would be to do  more of a hunter or huntsman costume.  There's no right or wrong here. 

 

Assassins should be more ninja-style, but if you want to class it up, go for the secret agent trench coat with the hidden earpiece.  Maybe carry a steal briefcase or violin case with a toy weapon inside. 

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Last, for any of these investigative roles - there's always the Scooby Doo costumes, Ace Ventura, Inspector Gadget, as they were investigators. 

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