Usually only the following vowel letters are relevant, and only for sc, not for sch.
Main possible pronunciations of sc: /sk/, /s/ or /ʃ/
In general, "c" in "sc" follows the usual rule for "c": /s/ before "e, i, y" and /k/ before "a, o, u" (the same rule of thumb applies to "g", which is usually /dʒ/ before "e, i, y" and /g/ before "a, o, u").
The /ʃ/ in "conscious" is because of the coalescent palatalization of /sj/ to /ʃ/. We may also see palatalization of unstressed word-internal /si~sɪ/ to /ʃi~ʃɪ/ when there is another vowel following, as in fasciation, which is pronounced either as /ˌfæʃiˈeɪʃən/ or as /ˌfæsiˈeɪʃən/ according to the American Heritage Dictionary.
The identity of the preceding vowel/vowel letter is normally not relevant.
Overview of the usual pronunciations:
- -sc- = /sk/ before a, o, u
- -sc- = /s/ before e, i (y) in most cases
- -sc- = /ʃ/ before e, i followed by another vowel letter (only in some cases)
The rules given here have exceptions. Some exceptions are optional, while other exceptions are less optional:
Some people pronounce proboscis with /sk/, but it's also fine, or even preferable, to pronounce it with /s/.
It seems to me that fascism is almost always pronounced with /ʃ/ nowadays, probably due to influence from Italian, where sc regularly represents /ʃ/ and not /s/ before i or e in any context. Still, you can find the pronunciation with /s/ listed in a few dictionaries, like Merriam-Webster and O.E.D. Suppl. (1933) (the modern Oxford English Dictionary mentions this, but doesn't list it as a modern pronunciation).
The well-known example of sceptic (and derived words sceptical, scepticism etc.) is invariably pronounced with /sk/, although in the United States the alternative spelling "skeptic" tends to be used, which corresponds more regularly to the pronunciation.
Main possible pronunciations of sch: /sk/, /ʃ/, /stʃ/ (rarely /s/)
The pronunciation of "ch" or "sch" is a quite different matter. The range of pronunciations for "ch" without preceding "s" is covered by How do I know when a word with "ch" is pronounced hard or softly?
Basically, there is no useful rule related to vowels for the pronunciation of sch. It can be pronounced /sk/, /ʃ/, /stʃ/, /s/ or even /ʃtʃ/ depending on various factors.
The pronunciation /stʃ/ can normally only occur before a vowel and in the middle of a word, not at the start or end. It seems to only occur in contexts where the /s/ and /tʃ/ can be analyzed as belonging to separate syllables, so perhaps a more detailed transcription of this pronunciation would be /s.tʃ/.
The consonant cluster /stʃ/ is sometimes pronounced as something similar to [ʃtʃ] because of processes of assimilation. If we set aside this arguably non-phonemic phenomenon, /ʃtʃ/ is a very uncommon pronunciation of sch. Some speakers may have /ʃtʃ/ in the word borsch, a recent loanword. Collins lists the pronunciation /bɔːʃtʃ/ along with alternative pronunciations /bɔːʃ/ and /bɔːʃt/. (Of course, in American English, a rhotic vowel like [o˞] or [ɔ˞] would be used). Alternative spellings of this word include "borshch", "borscht", "borsht".
The pronunciation /s/ is very rare. As far as I know, it only occurs in one pronunciation of schism (there is also a pronunciation with /sk/), and an obsolete pronunciation of schedule.
I think that schm is always pronounced /ʃm/, and schn-, schr-, schl- are always pronounced as /ʃn, ʃr, ʃl/ when they come at the start of a word/syllable. In word-medial contexts, there are a few very rare cases where -schn- might be pronounced as /sk.n/ (or /s.n/?) in scientific vocabulary (e.g. Ischnacanthus), or where -schr- might be pronounced as /s.kr/ (e.g. dyschromatopsia).
Etymology is important: broadly speaking, "sch" is pronounced /sk/ in words from Greek and /ʃ/ in words from German. Words from French are more unpredictable ("eschew" is from French, as is "schedule"). In words from Italian, where "sch" only is used before the letters "e" and "i", "sch" is pronounced as /sk/ (e.g. scherzo) ... except for when it isn't (as in one common, but commonly criticized, pronunciation in English of the word bruschetta).