Can a postdoctoral fellow, who holds a PhD in the relevant field, serve on a PhD student's dissertation committee? This might vary by field, but I'm curious whether there is any general trend on whether being a committee member depends on holding a PhD or holding a faculty position.
The main concerns for bringing in people to sit on a dissertation committee are satisfying the institutional regulations for committees (X members from outside the department, Y members from inside who aren't the advisor, Z members from outside the college) as well as bringing in possible expertise and exposure to additional ideas and approaches that may help the thesis candidate.
Normally a postdoctoral fellow will not count toward any of the composition requirements—one must be a faculty member or hold a similar position somewhere else (e.g., government research or industry). However, if one were to have a unique set of skills that someone thought it was essential to ask you to act as an additional member of the committee—this would typically be at another institution—there might be room and incentive to make something happen.
In the U.S., at many universities, in addition to being "on the faculty", one cannot advise PhD students or serve on the relevant committees without further vetting. At my current place, this is called "being a member of The Graduate Faculty". Peoples' credentials have to be submitted to some campus-wide entity for scrutiny.
... and it is rare (in mathematics, at my university) that post-docs go through that process, which takes several months, in any case.
From the other answers, it seems clear this varies greatly. I imagine it varies greatly by committee chair (preference) as well.
At my institution, post docs may serve as Technical Consultants with no departmental approval required.
They may also serve on committees, but this requires a vote from the department faculty to grant graduate faculty status. If granted, the post doc could then serve on committees or even co-chair a committee with a faculty member.
My university, which is in Australia, does not have dissertation committees. It has supervisors. Each student gets at least two supervisors. A new postdoc can hold up to 25% of responsibility for supervising up to three students. The rules are the same for temporary or permanent employees. Academics can earn additional supervisory responsibility with experience and training. These supervisors do not evaluate the students' theses. That is done by external experts.
This system is quite unusual, in my opinion. As the other answers say, at most institutions, postdocs cannot be on PhD committees.
This varies by country. I've known it to happen in UK, for example. In other places it would be forbidden, I assume. However, it is very unlikely that such a person would be the lead supervisor, at least on paper. But as a committee member it just depends on local custom and any applicable laws or regulations.
As for a general trend, I doubt that it is very common anywhere, nor likely to become so.
In an extreme case, a post doc might be, in fact, the main advisor due to specialized knowledge not held by other faculty. But a regular faculty member would almost certainly be the advisor of record and the main signatory on the dissertation. This might actually be seen as advantageous in some places to help the post doc get experience with the process. But the actual responsibility would remain, as I said, with a regular member. I'd be surprised to learn of any exception.