Silent <h> is a bit complicated as silent letters go. Most of the time it is used to represent long broad vowels in foreign words. I would treat these cases -- <ah>, <eh>,<oh>, and <uh> –- as consonant digraphs, though they could be analyzed as instances of silent <h>. My database contains 96 instances of this kind of <h>: aliyah, amah, ayah, ayatollah, baht, brahma, bremsstrahlung, cahier, challah, cheetah, chutzpah, dahabeah, dahabeeyah, dahl, djellabah, dourah, ephah, fellah, fellahin, foehn, fuehrer, galah, gerah, goombah, habdalah, haftarah, halavah, hallah, hallelujah, halvah, hamzah, haphtarah, hookah, horah, hosannah, houdah, howdah, hutzpah, ihram, jarrah, kaffiyeh, kiblah, kieselguh, kohl, loofah, maharajah, mahjong, mahzor, mashgiah, mastabah, matzoh, memsahib, menorah, messiah, mezuzah, mihrab, mikvah, mitzvah, mullah, mynah, nargileh, nullah, obeah, opah, pahlavi, pariah, pharaoh, prutah, punkah, purdah, rajah, rupiah, sahib, sahuaro, savannah, selah, shah, shari'ah, shibah, shillalah, shittah, shivah, surah, tabbouleh, tahr, tahseeldar, tempeh, torah, tussah, tzedakah, uhlan, verandah, wallah, whidah, whydah, yeshivah.
This same type of possibly silent <h> appears in nine words derived from proper names: abohm, boehmite, buhl, gahnite, graham, hahnium, nielsbohrium, ohm, teraohm.
And in six cases the <h> marks long broad vowels in more native words: buhrstone, demijohn, john, mahlstick, moolah, sirrah. This group would include a number of interjections: aah, ah, bah, blah, eh, hah, huh, hurrah, huzzah, oh, ooh, oompah, pah, pooh, rah, uh, yeah
But definitely silent <h>’s appear in words in which over the centuries the [h] sound has been lost in most persons’ pronunciations: annihilate, behemoth, coheir, dah, exhale, exhaust, exhibit, exhibition, exhilarate, exhort, exhume, heir, herb, honest, honestly, honesty, honor, honorable, hour, nihilism, prohibition, shepherd, vehement, vehicle. I suspect that these final 24 are the silent <h>’s you are most interested in.