Hatnotes which are composed of an indent (:) plus italic text are shown by mistake in link previews. However, the link preview correctly removes hatnotes which are made from the various hatnote templates, such as {{hatnote}}, {{About}}, {{For}}, etc.
Example: page – link preview
This is a response to CKoerner (WMF)'s reply on this topic.
From an editor-perspective, this could be seen as a good thing (or at least a beneficial aspect), because it helps us notice things which need to be fixed! Admittedly not in the cleanest way possible, but...
I've fixed that instance.
@Quiddity You're right, and I fix it every time I come across it, but it's how often it comes up that made me consider it a problem. I can't count the times I've had to fix this format of hatnote. For this reason, I was thinking it could be easier to configure a way to not show it in link previews instead, but of course it's up to the developers.
The answer here is definitely to fix the instances of incorrectly-formed hatnotes, rather than to hide the problem in any way. I've personally fixed thousands of these, and as far as I can tell this is really a problem that needs human effort to clean up: the badly-formatted cases are too inconsistent to cleanly handle automatically. Beyond showing up badly in previews, the bad :'' format also represents inferior formatting for accessibility.
I'd like to mention in particular some regex-based searches for these sorts of issues saved in my sandbox (permalink).
If we get more people together fixing them, we could reduce the problem significantly. If enough people are interested, we could even start a WikiProject to coordinate effort. I see hatnotes as a key part of the template infrastructure and have pushed a lot of effort into making them more functional and standardized (e.g. creating Module:Hatnote list, or umpteen TfDs to cut down on needless template sprawl).
Re: "we could even start a WikiProject [...]" - at Enwiki, this would best be handled by WikiProject Disambiguation. I suggest copying your great sandbox list, into a new sub-section at w:en:Wikipedia:WikiProject_Disambiguation#How_to_help.
(Also, kudos on your great "Notes from a survey [...]" table :-) Maybe that could become a WP:WPDAB subpage, or talkpage section, to enable collaborative analysis?)