The importance of credentials in pursuing an academic career in pure mathematics I would like to get some information and opinions of how the prestige of your so-called "credentials" helps or hurts in getting an academic job. I'm interested in this with respect to pure mathematics (I state this since I'm not sure if this is the same of all STEM subjects).

I guess that it is universally accepted the most important factor in one's academic career is his research. Of course, there are some more intricate relevant matters like a relationship with faculty of the department you apply to, success in teaching and the intersection of your research interests and faculty members' research interests. 

However, the I wonder how pure prestige of one's credentials matter. Let me state it more clearly:

1) The prestige of your graduate school. If we have two candidates for a postdoc/tenure track with relatively equal research experience, but one holds a PhD from Harvard, while another from University of British Columbia, or University of Pennsylvania, or something else good, but not as famous and "shiny" as Harvard (or MIT/Princeton, for that matter), would a Harvard alumni be in a better position?

2) The fame of your doctoral advisor. Again, I advise you to consider how much the pure "fame" of your advisor matters, all other things aside, like the fact that a strong mathematician could help you mature as a mathematician too.
Would having a Breakthrough Prize in Math/Fields winner boost on CV in way that math departments would more readily accept you as a postdoctorial researcher/assistant professor? Of course, this question is not only about Fields medalist, it was just an example.

3) The prestige of the institution where you did your postdoc. While points 1) and 2) were also related to getting a postdoc position, this matters only to getting an assistant professorship/tenure track. Usually, the advice for choosing a postdoc is to find a suitable researchers with respect to your interests. But this question again asks you to look on the matter from a different angle: the one of how being a postodoctorial researcher in a high-ranked institution assists in getting a tenure track.

Again, I wish to emphasize that this question is not about how a quality of a graduate school, a strength of a PhD advisor and a research atmosphere in a postdoctorial institution matters to an aspiring mathematician with respect to getting stronger as a researcher and a teacher. This much I know.
I wish to hear some opinions on more "petty" stuff like "shiny", "glowing" names of departments or advisors, or relatively less famous ones.
First, the statement that your research is the most important factor only applies to research universities.  Other academic institutions have, for good reason, other important criteria.

Second, the first thing you should understand about academic hiring is that the hiring committee is unable to evaluate with a high degree of accuracy your research.  The error bars on the evaluation of your research are very large, and, particularly for more junior mathematicians, the noise can be almost as big as the signal.

Third, for junior mathematicians, a significant portion of the evaluation of your research comes from your letters of recommendation.  The quality and significance of your publications is much more important than the quantity, and we all know the journal reviewing system is very noisy, with a very large difference between the best and worst papers in a given journal.  Furthermore, your most significant work may not even have had the time to go through the peer review process.  You might think that the committee could try to directly evaluate your work, but in almost all cases, no one on the committee will have the expertise to do so, and in the remaining cases, only one person on the committee will, and they still have to convince everyone else.

Now to actually saying something about your question:

The fame of your recommenders is generally thought to be a noise reducing factor in evaluating your letters.  If a mediocre mathematician thinks you are good, who knows?  They aren't good themselves, so what do they know about being good or not?  If a well-known outstanding mathematician has worked with you and thinks you are good, then it seems more likely that you are actually good.  (In reality, I'm not sure how well ability in mathematics correlates with ability to judge mathematical ability in others, but I'm talking about how search committees tend to think.  There is at least some positive correlation, in that people who aren't mathematicians almost certainly don't have anything useful to say first hand.)

The places you do your PhD and postdoc(s) also likely put some floor on your ability, so they are also noise reducing factors.

Generally speaking, noise reduction on the bottom end helps your candidacy.  So going to a strong department and having a well known advisor doesn't just make you a stronger mathematician; it also makes you look like a stronger mathematician, or, at least, look less likely to be merely average or good.

On the other hand, search committees also recognize that people who went to less prestigious departments and had less capable advisors have had less opportunity, and more of their success can be attributed to them alone.  This might lead the committee to think the candidate is more likely to be successful in the future.  This means someone from a weaker program who can be a convincingly strong candidate might have an advantage.