What event caused the start of the modern era?

The answer is "it depends what you specifically mean by modern era".

Depends on how one defines the "modern era". The Industrial revolution is what many consider to be what began the modern era of most European societies, but few people agree on an exact date when that revolution became manifest. Some connect that loosely with the revolutionary scientific discovery of Antoine Lavoisier in France, which proved, among other things, that alchemy was an impossibility. Others point to events around the American revolution, or the "Glorious Revolution" in England.

Still other opinions say that we only became truly modern with the advent of the atomic age or even the space age in the 1950s and '60s. In artistic terms, the end of World War 1 is used in western art and music as the general point after which artists are referred to as modern. In religious terms, however, opinions for the most part go much further than that. Modern Rabbinic Judaism, for example, usually refers to the development of the religion since the compilation of the Talmud, around the 6th Century CE.
In paleontological terms, "modern" could refer to the period of recorded history (up to about five or six thousand years ago), or up to the earliest specimen of Homo sapiens being found in Africa and the Fertile Crescent (up to 100,000 years agoor more).

So really, the term "modern" depends on the discipline one is referring to.

Like the answer above, I don't think we can point to any single event and universally agree to call it the start of the "modern era". It appears the modern era has different starting points from different historians, and depending on exactly which starting point you choose, you can identify an event to kick it off.

Wikipedia defines the "modern era" as:

Modern history, the modern period or the modern era, is the global historiographical approach to the timeframe after the post-classical era in European history (known as the Middle Ages).

https://en.wikipedia.org/wiki/Modern_history

The early modern period is "variously demarcated by historians as beginning with the Fall of Constantinople in 1453, with the Renaissance period, and with the Age of Discovery (especially with the voyages of Christopher Columbus beginning in 1492, but also with Vasco da Gama's discovery of the sea route to the East in 1498)"

https://en.wikipedia.org/wiki/Early_modern_period