Let me see / Let me think - Used to buy time to consider something.
Let me know - Used to ask for information or updates.
Let me handle this - To take responsibility.
Let me introduce...: - Used to make introductions.
Let me rephrase - To explain differently.
Let me know if... - Offering flexibility.
Let me take a look - Offering to examine something.
Let me fix that - Offering to resolve an issue.
Let me be/Let me alone - A request to stop being bothered or to be left alone.
Let go - To release physical hold or to dismiss an employee.
Let alone - Used to emphasize that something is impossible, much less something else