There are really 4 kinds of Munach: Legarmeih, Munach, Ilui, and Mekhurbal (AKA Karbalta). Legarmeih is disjunctive (pausal), and the other three are conjunctive (connective).
Legarmeih is always written with a vertical line after it. Unfortunately, that line serves two purposes in Tanakh: 1) indicating a short pause after a conjunctive note (usually called a "Psik/Paseik"), 2) indicating a preceding note that has multiple meanings is of the disjunctive type (sometimes itself called "Legarmeih"). Type two is used to distinguish Legarmeih from the others (as well as Shalshelet Gedola, Azla Legarmeih, and Mahpakh Legarmeih from their conjunctive doppelgängers in the three poetic books).
How can we identify a disjunctive Legarmeih from a conjunctive Munach which has a pause after it? One giveaway is if the "Munach" in question is preceded by a Merkha. Then we know it's a Legarmeih because Merkha is the conjunctive note which serves it (like how Tevir is served by Darga). But that doesn't always work.
Thankfully, the Masoretes were aware of this issue and included lists in the Masorah of every Psik in Tanakh (eg. in the Leningrad Codex after each of T N and K). If a line is not on that list, it must be a Legarmeih. So problem solved. The same technique can be used with the other notes above in the poetic books.
As it happens (and this mnemonic is noted in many of the old grammatical works cited below and is accepted by Wickes), if you look through the lists you'll find that a "Munach" with a line is a Legarmeih whenever it precedes a Revi'i (except Isaiah 42:5) plus a handful of other times before other notes.
A masoretic note to Isaiah 42:5 in the Leningrad Codex confirms this, by noting explicitly that the line there is a Psik. A masoretic note in the Aleppo Codex to Joshua 9:12 also confirms this, as it notes the opening word appears thrice at the start of a verse with that Trop. This only makes sense if it is a Legarmeih not a conjunctive Munach, as there are more than three places that that word opens a verse with a Munach.
(R Mordechai Breuer notes that in the 3 poetic books the parallel of Legarmeih-Munach-Revi'i is MahpakhLegarmeih-Munach-Tzinnor and again almost (exceptions Tehillim 68:21, Mishei 1:22) nowhere in those books do we find Munach-Psik-Tzinnor, yet MahpakhLegarmei-Tzinnor is common. A single Mesharet with a Psik apparently naturally converts to the Legarmeih.)
Now, regarding the conjunctive ones:
An Ilui serves three notes: Etnachta, Zakef, Zarka.
A "Munach" just preceding an Etnachta is an Ilui (eg. Gen 1:1). If the Etnachta has multiple "Munachim" only the last is an Ilui (eg. Gen 40:16).
A "Munach" just preceding a Zakef is an Ilui (eg. Gen 1:2) if it is not on the first letter of the word. If the Zakef has multiple "Munachim" only the last is an Ilui (eg. Gen 3:12).
A "Munach" just preceding a Zarka is an Ilui (eg. Gen 1:7). If the Zarka has multiple "Munachim" only the last is an Ilui unless the words are all monosyllabic when all are Iluyim (ex. Exo 4:11). "Munachim" preceding a Telisha Ketana which serves a Zarka are also all Iluyim (eg. Exo 17:6) according to most, and some also include the "Munachim" between a Zarka and Segol. (Disclaimer: the rules of which notes serve a Zarka in general are complicated and subject to disputes and exceptions and disputes about exceptions, and this aspect of them is no different.)
A Mekhurbal is found serving a Zakef either as the first of two "Munachim" (eg. Gen 3:12), or as the only "Munach" if on the first letter of a word (eg. Gen 1:30).
Any other "Munach" (such as before a Pazeir, Gershayim, Revi'i, Darga, Mahpakh, etc.) is a Munach (unless it's a Legarmeih, as above).
At this point most of you are probably wondering if I'm making all this up, but you can see these rules discussed in classical grammatical works like Ibn Bilam's, Rabbeinu Tam's Piyut, Tuv Ta'am, Lechem HaBikkurim, Darkhei HaNiggun, Eit Sofer, Binyan Shelomo, Siach Yitzchak, the Tijan, Mishpetei HaTa'amim, and more.
So what do they say, you ask? Just look at their names: an Ilui goes up, a Munach stays down, and a Mekhurbal is something like wrapped or twisted. You already probably usually do an Ilui in the sequence "Pashta-Ilui-Zakef" and a Munach in the sequence "Munach-Revi'i". The Mekhurbal is probably the fancier sound you sometimes hear read for a "Munach" before a Mahpakh. (While according to the above that would be a mistake, Binyan Shelomo actually suggests that the tune there proves that a "Munach" before a Mahpakh is also a Mekhurbal not a Munach, but this is just his speculation and goes against the traditional grammarians.)
Now all you need to do is to sing the right ones in the right places!