I am working on drawing the syllable structure for the word crazy. So far within kreizi, ei and i are nucliet, kr is an onset, but I am stuck on the 'z'. There are many words that start with z in the English language so I am inclined to say it is an onset as onsets are greedy.
Nobody has devised a satisfactory test for syllabification especially in English. The literature is chock full of contradictory claims as to the syllabification of words, all supported with a bit of evidence. There are some undeniable facts, such as that the word big begins with a labial. There are some facts whose analysis is theory dependent but the variation is limited, for instance big is said to begin with a voiced stop (or you could use the term plosive, or you could say "unaspirated" or "lenis", but not "nasal" or "ejective"). Not only are the criteria for syllabification judgments highly variable, the set of structural entities posited by prosodic analysis is extremely variable. The very existence of an object "syllable" is contentious. In Strict CV phonology, there is no representational object "syllable", nor "onset" as distinct from "coda", instead there are Cs and Vs and lateral relations between Cs and Vs. Even Clements & Keyser's original CV phonology did not have objects "onset" vs "coda", it simply assembled Cs and Vs under a syllable node.
As for English, consonants in the context ˈCVCV have been subject to very many analyses. One hypothesis, popularized by Kahn 1975, is that consonants are attracted to a preceding syllable which ends with a vocoid (vowel, glide or r in some dialects), thus the consonant is both in the first syllable and in the second (it is "ambisyllabic"). An alternative hypothesis is that the consonant is exclusive in the preceding syllable therefore the second syllable begins with a vowel. Whether or not that relates to "onsets" or "codas" depends on whether you also have such nodes in your representational arsenal (and what is your theory of what onsets or codas can be or do?).
There seems to be agreement that in hanger, the consonant [ŋ] has to be in the first syllable, because words cannot begin with [ŋ] in English therefore syllables cannot begin with [ŋ], therefore [hæ.ŋɹ̩] would be theoretically impossible, therefore it must be syllabified [hæŋ.ɹ̩]. That kind of argument does not help with "crazy".
There is a line of reasoning to the effect that "busy" must be syllabified [bɪz.i], because no word in English ends in [ɪ] (or any short / lax vowel), therefore syllables cannot end in a lax vowel, therefore "busy" must be syllabified [bɪz.i]. But "busy" is different from "crazy" in the vowel quality of the stressed syllable. There is also the matter of aspiration and flapping, where we often say that /t/ is aspirated in syllable-initial position. But there is no aspiration of /t/ in "crater", therefore /t/ cannot be sylable-initial, or (following Kahn's analysis) exclusively syllable-initial. Of course, /t/ is not /s/ so that evidence does not directly prove or disprove a particular analysis of crazy. Plus, there is an alternative statement of aspiration and flapping in terms of initial position in the stress foot, and it is generally agreed that /t/ in crater is not foot-initial.
In short, yes or no, depending mainly on your theoretical premises. It is almost trivial to identify a theoretical premise that precludes z as onset or as coda in this instance, it is totally non-trivial to prove that that theoretical premise is correct.