[verb た-form] + 拍子に [verb た-form] + 弾みに [verb た-form] + 弾みで
た拍子に and た弾みに say that the moment one thing happened, it set off something nobody intended. The verb before them is in the た-form and is usually a sudden movement – standing up, turning round, sneezing, tripping – and the second half is the accident.
Because the result is an accident, the second half can't be something the subject chose to do. It often ends in てしまった.
The two are mostly interchangeable, and both are at home in conversation as well as writing. はずみ弾み is "bounce, momentum", so 弾みに suits a physical jolt – a fall, a collision, a sudden stop – and also appears as はずみ弾みで. ひょうし拍子 is "beat, timing" and puts the weight on the instant.
×外に出た拍子に、雨が降り出した is wrong: going outside did not make it rain.
After a noun, 拍子 is mostly met in the set phrase なんか何かのひょうし拍子に, "for some reason", "somehow".