In patent search, there is no "search" box. Instead, the "quick search" forces you to specify two (and exactly two) text queries on the database columns with obligatory boolean operation.[1] Even if you happen to find an interesting patent, good luck linking to it (which should be the number one service they provide - GET individual patent documents). The page showing the patent document has a dozen cryptic query parameters in the URL, some of which relate to the search query you used to find the patent! No "make shareable link" button to be seen, either.
And don't get me started on the trademark search, or Trademark Electronic Search System (TESS), as they like to call it.[2] When you navigate to the front page, you get a private session identifier - in your URL of course! And when you search for a trademark ("simple search" is intuitively known as "New User" here) and select a TM to view, you would be excused of thinking that the short URL in your browser address bar is the linkable URL of this entry. But no - it's just your session identifier along with the document's index in the results of your last search query.
When you leave the trademark site or just click "Logout" (since you're a kind person - they after all ask you "logout when you are done to release system resources allocated for you"), that URL is gone in the wind. If you shared a link to that trademark to your friend, they only get this very helpful page:
<H1>This search session has expired. Please start a search session again by clicking on the TRADEMARK icon, if you wish to continue</H1>
So no way to link to individual TM registrations here either.