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I have some UNC notes that I think will sell for around $40. What is the best way to ship/pack them?

Put it in a standard plastic currency sleeve inside a standard envelope, then put that in a USPS photo mailer for additional stiffness. Or if it is a stack of them you might have to be more creative.

You can use a manilla envelope if you have a stack of them and they are in currency sleeves. Put them in the envelope then either fold it in half or thirds depending on the size of the notes. Then you can just tape it up and mail it.

Currency sleeve, regular envelope and maybe a piece of cardboard for some support. I really like shipping in usps flat rate envelopes, offers some support over a manilla envelope and also is free and provided by usps.