I'm not sure if this is intentional (it seems to be), but upon pressing "Search" on any Special:Search page with profile=advanced in the URL (e.g. https://commons.wikimedia.org/w/index.php?title=Special%3ASearch&profile=advanced), the search will be sent as a POST request instead of a GET request, and the URL will become the index.php page (e.g. https://commons.wikimedia.org/w/index.php). As a consequence, it's not possible to copy the search URL, even though the search is performed.
Using Firefox's element inspector, it seems that changing <form ... method="post"> to <form ... method="get"> is sufficient to make the search parameters appear in the URL. Is there any particular advantage to sending these searches as POST requests? (I would note that this is actually inconsistent, since if you switch on the advanced search parameters at Special:Search the next search is still sent as a GET request, so initially the advanced search parameters do appear in the URL, but the search after that one is sent as a POST request.)
Normal search (archive):
<form id="search" method="get" action="/w/index.php">
Advanced search (archive):
<form id="powersearch" method="post" action="/w/index.php">