Both "zijn vergeten" and "hebben vergeten" are correct in present perfect, but they emphasize different aspects. "Hebben vergeten" focuses on the action of forgetting (e.g., a fact or to do something), while "zijn vergeten" highlights the resulting state (e.g., leaving something behind).
In practice, the choice depends on what you want to stress. When you forget a fact or that you had to do something, "hebben" is typical: "Ik heb zijn naam vergeten" (I forgot his name). When you forget to bring a physical object or leave it behind, "zijn" is common: "Ik ben mijn telefoon vergeten" (I left my phone behind). For forgetting to perform an action with "te" plus an infinitive, both occur, but "zijn" is the default in daily speech: "Ik ben vergeten te antwoorden" (I forgot to reply).
The tradeoff is subtle. Using "zijn" often implies the thing forgotten is left somewhere or omitted in a tangible way, while "hebben" feels more mental or abstract. In some regions or formal writing, "zijn" with vergeten can sound slightly dated; however, it remains standard in the Netherlands for the "left behind" sense. The verb "vergeten" itself is exceptional because it can take either auxiliary depending on context, unlike most other verbs.
To build confidence, look at example sentences in online conjugation tables or learner dictionaries and note the auxiliary used with the specific meaning. Practice by describing your own forgetfulness in Dutch, deciding whether you left something (zijn) or simply didn't recall (hebben). Over time, the nuance will become intuitive. Resources like grammar textbooks or reliable apps explain this as part of the broader "zijn vs hebben" rules for perfect tenses, but for vergeten, context is king. There is no single right answer for every situation, so embrace the flexibility and pay attention to how native speakers use it in conversations.