syllabic plosives

Timothy Birkett wrote:

I was put on the spot the other day when a student asked me if /p/ in potato could be marked as a syllabic consonant - assuming there was no audible schwa after it. Thinking that p couldn't be the nucleus of a syllable because it's not sufficiently sonorous, I said "no". Is this true, and if so, how would you transcribe it?

I agree that the answer is “no”, and I would transcribe it straightforwardly as p: for example, hæv əpˈteɪtəʊ.
In principle, I would say that plosives can never be syllabic. People have sometimes tried to claim that a very reduced form of thank you, kkju, has two syllables, the first consisting of an inaudible and unreleased syllabic k. And in Romanian and Japanese there are high vowels which become devoiced or disappear after voiceless plosives, giving contrasts such as Romanian lupi̥ vs. lupu̥ (or better lupʲ vs. lupʷ) — but I would say that these original disyllables were reduced to monosyllables by this process.
Nevertheless I agree that there is a fine line between devoicing a vowel and deleting it entirely, perhaps leaving behind a secondary articulation on the preceding consonant.
I have often had to correct beginners who wanted to transcribe wanted, for example, as ˈwɒntd̩, imagining that the final sequence ɪd or əd was a syllabic d. But it isn’t.