There doesn't seem to be any pattern to when it's "Captain's Log, StarDate xxx' and 'Captain's Log, Supplemental". Is there any?
"Captain's Log, Stardate XXXX" is the beginning point of an entry, and generally occurs at the start of a new episode or adventure.
"Captain's Log, Supplemental" is a later addition to an existing log entry, generally made mid-way through an episode or adventure. The idea is that the Captain came back later and tacked-on another bit of important information to an entry they had made earlier.
In my experience, the pattern holds up quite well:
In-universe, each new log entry is for a major event or mission, and supplementary material is added during the mission or after the mission is completed.
Out-of-universe, each new log entry begins a new episode, and supplementary entries are usually a way of either catching up viewers who may be starting the episode mid-way through, or quickly establishing events that have occurred off-screen during a commercial break.