This is one of the topics addressed by Mike Brame in his MIT dissertation Ch. 5, for Classical Arabic, however I have to say that I find his discussion inconclusive.
The prosodic pattern of verbs and deverbals (CCVC, CVCVC, CVC:C...) is convincingly reducible to non-lexical factors (e.g. "is this is 2nd measure derived verb; is this perfective, or imperfective"), and the vowel selected within the root is largely predictable from such morphological factors. However, roots still fall into lexically-governed subclass w.r.t. selection of vowel quality. There are, again, various assimilations of vowels to adjacent consonant (glides and gutturals). In his scheme, roots underlyingly have the form /CaCVC/ where /a/ is a fixed vowel in all verbs (he does not actually give an argument for /CaCVC/ rather than /CCVC/, he simply observes that the derivation CV+CaCVCVC→CV+CCVCVC follows a simply phonological rule). The quality of the second stem form is lexically determined, thus underlyingly present. The important conclusion which isn't in doubt is that there are lexically-determined sub-classes of verbs e.f. qatal-at, ta-qtul-u 'kill', nazal-at, ta-nzil-u 'descend', labis-at, ta-lbas-u 'dress', kabur-at, ta-kbur-u 'become big'.
There are two general approaches to this distinction, the adjudication of which depends on your theory of phonological analysis. One is to assign lexical diacritics A, B, C, D to roots and devise rules that say that A-subclass verbs have [a] in the perfective and [u] in the imperfective, and so on. The alternative is to analyze the vowels so that the underlying vowel determines the surface vowel, given appropriate phonological rules. Complete phonological predictability is not possible, given non-alternating roots like /kbr/, but Brame suggests that this subclass may be predictable since it largely (or entirely?) is composed of stative and inchoative verbs. His analysis posits an arbitrary lexically-distributed diacritic, which is a consequence of the SPE-style analysis that he gives: the implication is that this diacritic is not a dictionary property, it is predictable (but not be a proper phonological rule).
The one-way predictability of the ablaut relations suggests that the underlying vowel is that of the perfective (/i,u/ → a; /a/ → [i]), but Brame rejected this analysis in favor of a "perfective-to-imperfective" mapping with an arbitrary split in /a/ which becomes [i,u] unpredictably. The argument is rather complicated, but hinges on the pattern of lame (glide-final) stems with /w/. The appearance of [y] in [ta-rḍay-ā-ni] 'they f.d. become content' from /rḍw/ is a seeming puzzle since nothing would tell you that w→y/a_. But it is established that w→y/i_, thus
/ta-rḍiw-ā-ni/ → ta-rḍiy-ā-ni → [ta-rḍay-ā-ni], where the last stem is the Ablaut rule – under the assumption that the underlying vowel is the perfective vowel, not the imperfective vowel. The disadvantage of this analysis is that arbitrary diacritics seems to be necessary to say which cases of perfective /a/ become [i] and which become [u]. His solution is to posit an underlying distinction between /a/ and /æ/.
From a contemporary theoretical perspective, the evidence does not come down strongly in favor of a "no vowel" analysis versus the "some specified vowel" hypothesis. Aided with an appropriate analytical dictum, you can discern an argument favoring one or the other of these hypotheses, it's just a matter of picking the right dictum, at least for Classical Arabic. Vowel choice in verbs is partially lexical, but what that says about underlying forms is up in the air. This might be a good time for theorists to revisit the Classical Arabic root-vowel problem.
