Hi Alexey Those two terms are interchangeable, meaning nobody will judge you for using either. Generally speaking though, 'happen to' refers to a physical person or object, while 'happen with' could refer to something more abstract. For example: "What will happen to John, when he leaves?" "What will happen with the weather, if the ice caps melt? (Weather in this context referring to a state or condition​, which is abstract)

Hey Alexey, The most common is "happen to", although "happen with" wouldn't be grammatically incorrect. Some say that "happen to" refers to when something happens to somebody or something, it takes place and affects them. i.e: Someone got hit by a car, then something happened to him. And "happen with" is more used when something is going on with someone. i.e: Mark looked very tired today. What happened with him? I can't find a liable resource that proves it though, it just feels this way. But "happen to" is definitely more used.