I trim my images exactly; there is no border. I lightly sign the back of each print with soft pencil before dry-mounting it to a cotton rag board. Yes, the signature is then not visible, but could be detected with infrared or x-ray. The print and the board to which it is mounted comprises the entire work of art as far as I'm concerned, since I choose mat color and quality depending on the image. So, yes, I sign the mat board, in pencil, just under the bottom-right corner of the image. The window in the overmat is cut to allow a border of the undermat to show, along with the signature.
I also stamp the back of the mat board in the upper-right corner with a rubber stamp and using archival ink. The stamp says "Photograph by Doremus Scudder" and has places to enter the title, location and date as well as the print number. These I enter with pencil. I could use ink as well, I suppose.
This is a classic way to present art photos and is used by many. I know there is a trend to avoid dry-mounting, but I prefer the exact borders I get this way and the flatness it provides. Using cotton-rag board and archival dry-mount tissue along with proper display and storage should ensure maximum longevity.
Best,
Doremus