Starting in the mid-1950s, plastic straws began to replace paper versions at warp speed, and fast-food culture played a fundamental role. Plastic straws were cheap, durable and they would not tear apart as you pushed them through the crosshairs of to-go cups, also made of plastic. Still, accidents will happen, as seen below.
In terms of the modern era, at least, we seem to have come full circle, as paper straws are now touted as a sustainable alternative to plastic straws. There's plenty to be said about that on a scientific and technological level, but that is an article for another day.