Now if no one minds--
TLDR: When in doubt, Michael is your archangel of the hour. The rest work in shifts.
If anything here is wrong, someone with more knowledge can correct me!
What exactly is an 'Archangel'?
So, our first hurdle here is "what". And yep unfortunately our answers are already gonna be "well, it depends".
The earliest reference to the actual TERM "Archangel" is from "De Coelesti Hierarchia" by Pseudo-Dionysius the Areopagite (circa. 5th century AD). This is where we get the 'hierarchies of angels' for the Christian canon (at least, its beginnings).
Except Archangels are the second-lowest level of angel according to this hierarchy, only surpassing the generic angels themselves. So, why do we consider them the highest? A mix of the name (as it means 'chief angel'), and everyone's "favorite" poet, John Milton. And a wonderful little thing called "translation induced confusion".
So, what's the confusion?
When we look at the text of the Book of Enoch - everyone's favorite apocalyptic Jewish text - is where we get mentions of the highest ranking angels (including Michael, Raphael, Gabriel, and Uriel) with the title of "prince". Other angels with this title are Raguel, Saraqael, and Remiel. In english, they are referred to most often as "archangels" hence why we have this conundrum.
Alright so how MANY are there?
This depends on tradition. Within the bible proper (note: only in the christian canon), only Michael is used in reference to the term. So the rest is just - all over the place.
For (most) Catholics, it's only 3. Michael, Gabriel, and Raphael. While not part of the official canon, Uriel and Jeremiel are popular as well.
Within Eastern Orthodox there's technically thousands but only seven are venerated by name. Michael, Gabriel, Raphael, Uriel, Selaphiel (or Sealtiel - same angel, different spellings), Jegudiel (or Samuel - very different names but same iconography), Barachiel, and sometimes just to make it eight and really fuck around is Jeremiel.
In Coptic Orthodox we have Michael, Gabriel, Rephael, Suriel, Sadakiel, Sarathiel, and Ananiel.
In Ethiopian Orthodox, we have Michael, Gabriel, Raphael, Uriel, Phanuel, Raguel, and Ramiel.
The Protestant bible only includes Michael and Gabriel.
Anglicans however, recognize Michael, Gabriel, Raphael, Uriel, and Jeremiel.
Lutherans focus on Michael and Gabriel, with Raphael sometimes showing up.
Jehovah's Witnesses and Seventh-day Adventists have a fun "Michael the Archangel is actually Jesus" viewpoint. The LDS Church meanwhile only views Michael as one, but also that he was Adam. And Gabriel was Noah. No one knows who Raphael was but apparently he was also important. Literally no one else but these guys say this.
In Islam, we have Michael (Mika'il), Gabriel (Jibra'il), Raphael (Israfil - the only one not named in the Quran), and Azrael (Azra'il). Nice and simple.
As mentioned earlier, from the Book of Enoch we have Michael, Gabriel, Raphael, Uriel, Sariel, Raguel, and Remiel.
In the Apocalypse of Moses, we have Michael, Gabriel, Raphael, Uriel, and ... Joel. Who might be Metatron.
In the Kabbalah, we have far too many. Oh yeah here's where the real rabbit hole begins. There's traditionally twelve, including our main reoccuring ones, but guess what. That's right. We also have another eleven or twelve that share similar associations. Some of the traditional twelve ALSO overlap. It's fine. I would need a whole other post to go into the ones here.
Metatron is also here, sometimes. He continues to be an odd one.
So what's the final count (rhoughly)
Everyone Else - 1 (give or take)
(Look Tumblr has limited colors so those three have to share).
So the final say is generally : Michael and Gabriel are almost certainly both in there, Raphael is common, and Uriel shows up enough. So just have fun with it and don't worry about it too much because it's going to be like arguing translations when both of you don't speak the same language.