You can trade in your Wild Stamp in exchange for a specific stamp you want. However, they have to be of the same type:

Regular Stamps = Regular Wild Stamps
Gold Stamps = Gold Wild Stamps
Diamond Stamps = Diamond Wild Stamps

Take note that the star rating of the Wild Stamp should be the same as the stamp that you want. If it's not, it would not show up as an option.

Also, if you already have the stamps that are shown, you can use the Wild Stamp to trade for duplicate stamps or stamps you already have.
You must just  untick the option (located in the lower left of the screen) [SHOW ONLY STAMPS I DON'T HAVE]. If you untick this option, you would be able to choose for stamps you already have.