The word "syllabic" as applied to a letter that generally represents a consonant generally means "forms the peak of a syllable". Tashlhiyt (Berber) is said to have many syllabic consonants including /s̩ t̩/, which could in principle include syllabic glides. It is however recognized that this is not a fact of lexical representations, it is a fact about relatively superficial phonetics. At that level, /j w/ do have syllabic variants, which however are usually transcribed with the letters [i, u].
In a very few languages, so-called syllabic consonants have a derivationally deeper pedigree, resulting in minimal pairs, such as Swahili [mbuni] "ostrich" and [m̩buni] "coffee tree". Such contrasts exist in a number of Bantu languages, for example Swahili and Hehe. Even then, though, this is the result of a rule reducing /muC/ to [m̩] ("coffee tree" is /mu-buni/). This is the fate of so-called syllabic consonants in general, until you get to some of the challenging fricative vowels of Chinese languages such as Nantongese which is analyzed (by Ao) as having syllabic /ʒ̩ ʒ̩ʷ z̩ β̩/. In these languages, it is usually a matter of analytic premise that one should treat [z̩] as /z̩/ because there is no direct evidence that [z̩] derives from something else (e.g. /zə/). However, various theories provide indirect "evidence", in that they may prohibit syllabic consonants in underlying forms (typically by saying something to the effect that "the property that defines being a syllable peak does not exist in the lexicon").
The most compelling evidence for "syllabic glides" at a level more abstract that surface transcription would be a minimal pair, as we find in Swahili and a few other Bantu languages. Convincing minimal pair involving [u] and [w̩] or [i] and [j̩] are so far lacking. However, /j w/ do appear as uncontroversial [i u] in Tashlhiyt (and other languages) is exactly the contexts where you find other syllabic consonants. It is a classic observation of phonological theory and theories of features that glides and vowels are "the same" except for one property, the feature "syllabic" (or whatever one calls it).
