I have a book with a signature on the back of the front cover. Is it postable?

Yes, signed copies (even on the title page) as well as gifted copies with a msg to the recipient are acceptable.  Writing in general on inside of covers or blank flyleafs is acceptable.   BUT NO WRITING on text pages is allowed.

Read the help docs for the textbook exceptions.

You can look in the Help Docs to see what is and what is not acceptable to post for trade here.

Place your cursor over the Help Center tab in the silver bar near the top pf any PBS page, then select Browse Help Docs from the drop down. Click on Posting Books in the yellow bar at left, and read the topics "Is my book okay to post here?" and "What can be posted for swapping here?".

In response to Schuylar's specific question the answer is
Books that have a signature/note on the flyleaf or title page or inside front or back cover of the book are allowed

It is highly recommended that new members read the help documents to learn what is and what is not acceptable for posting. This really helps to prevent problems that result from posting items that are not allowed and from posting books incorrectly.