Happen, chance, occur, befall, betide, transpire are comparable when they mean to come to pass or to come about.
Happen is the ordinary and general term and may imply either obvious causation or seeming accident, either design or an absence of design; in its simplest use the term takes the event, situation, or circumstance as its subject, but it may take the impersonal it or the anticipatory there as its subject.
In still other phrases happen may take a person as the subject, especially when the verb implies a coming upon someone or something, or a coming into a place more or less casually or accidentally, rather than a coming to pass.
Chance is closer to happen in its idiomatic uses than any of the other words, and it too is found occasionally with the event as the subject; however, it differs from happen in uniformly implying absence of design or apparent lack of causation.
Occur in distinctive use carries an implication of presenting itself (as to sight, to consciousness, or to one’s thoughts); it is, in general, interchangeable with happen only when a definite event or incident or something that actually takes place is the subject.
Consequently occur is preferable to happen in negative expressions when the idea of presentation in the realm of fact is uppermost. This fundamental implication of presentation to sight or mind allows occur, unlike happen, to be used to suggest a coming to one’s mind or a meeting one’s eyes or ears (as in print or speech) or a turning up or appearing.
Befall and the less common betide are used in preference to any of the preceding words, especially in poetry or in literary prose, when there is an implication of a superior power determining events or of the lack of human power to foreknow or forestall them.
Transpire, which basically means to give off or escape in the form of a vapor, retains this notion in much of its extended use, in which it implies a leaking out so as to become known or apparent, but through a semantic shift the term has developed a value, disapproved by some rigorous purists, in which it is interchangeable with happen or occur.