We study the discrete bin covering problem where a multiset of items from a fixed set $S \subseteq (0,1]$ must be split into disjoint subsets while maximizing the number of subsets whose contents sum to at least $1$. We study the online discrete variant, where $S$ is finite, and items arrive sequentially. In the purely online setting, we show that the competitive ratios of best deterministic (and randomized) algorithms converge to $\frac{1}{2}$ for large $S$, similar to the continuous setting. Therefore, we consider the problem under the prediction setting, where algorithms may access a vector of frequencies predicting the frequency of items of each size in the instance. In this setting, we introduce a family of online algorithms that perform near-optimally when the predictions are correct. Further, we introduce a second family of more robust algorithms that presents a tradeoff between the performance guarantees when the predictions are perfect and when predictions are adversarial. Finally, we consider a stochastic setting where items are drawn independently from any fixed but unknown distribution of $S$. Using results from the PAC-learnability of probabilities in discrete distributions, we also introduce a purely online algorithm whose average-case performance is near-optimal with high probability for all finite sets $S$ and all distributions of $S$.