This is a broad category with tons of flexibility. Depending upon your assigned character role, you'll have a big range of choices for any of them, which should be fun for you.  What 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 wear simple business casual clothing with a fedora or deerstalker cap or go all in with a Sherlock costume. Another alternative is 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 bandanas and biker boots, opt for a spy trench coat theme, or 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. You could also 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.