As a guilty user of the Israeli mivta, I naturally am not naturally disposed to know the correct way to niqqudize any particular word in Hebrew. I've found that historical Hebrew phonology has been very useful to understanding how the whole system works. Anyway, I felt like it might be interesting to talk about a particular aspect of niqqud that I've recently been thinking about. If this is boring, I apologize.
To simplify a complicated discussion, Hebrew went through a phase where vowels could be short or long*. At some point this length was lost, but by then long /a:/ was pronounced further back, as the vowel /ɔ/**. The Masoretes wrote /a/ with patah (אַ) and /ɔ/, reflecting older long /a:/, with qamatz (אָ).
A general rule in older Hebrew was that in open syllables*** /a:/ replaced /a/. Thus in open syllables patah is rarely found. However there are some interesting exceptions:
1) Words with hataf vowels, e.g. רַחֲמִים = ra-ha-mim, where the first syllable ra has a patah. It's thought that the hataf patah added in forms like this was a late addition: at a late stage shva nah under gutturals (like ה ע ח א) became a hataf vowel. This is also found in words with other vowels like צהֳרים, צהֳלה, יחֱרד
2) Segolates like נער, בעל, נחל (= na-hal where the first vowel is patah). The dating on these is, I believe, tricky. However it's known that originally segolates only had one vowel, e.g. *malk > melekh מלך, or similarly *nahl > nahal. Thus the first vowel was originally in a closed syllable, so this makes some sense.
3) Words like ירַחֵם = ye-ra-hem where the second vowel is a patah. In cases like this the guttural was originally doubled (ye-rah-hem), but gutturals lost gemination (= doubling) at a late stage. Thus we see in other piel verbs like ידבּר there is a dagesh in the second root letter because it was geminate (ye-dab-ber)
Are there any other cases that I have missed? Please let me know if you're aware of any. Also, I apologize for the ad-hoc transcription; I figure that it does not distract from my point, but if it's distracting I can use something more precise.
* This is roughly comparable to the difference between the vowel in British lot versus start, where the second is more drawn-out.
**This is the vowel in the word caught, for speakers who pronounce it differently than the word cot.
***An open syllable ends in a consonant, while a closed syllable ends in a vowel. Thus הברה Ha-va-ra has three open syllables while יד yad has one closed syllable.
Main source: Phonology and Morphology of Biblical Hebrew by Joshua Blau (2010)