What counts as appropriate dress

The Vatican Museums state plainly that "entry to the Vatican Museums, the Sistine Chapel, St. Peter's Basilica and the Vatican Gardens is permitted only to appropriately dressed visitors." In practice, that means avoiding sleeveless tops, low-cut necklines, shorts or skirts above the knee, and hats.

What to bring if you're not sure

A light scarf, shawl or long shirt to throw over bare shoulders covers most situations, and is easy to carry on a hot day when you'd otherwise be tempted to wear a tank top. The same applies to a knee-length wrap for shorts or skirts that run a little short.

Is it strictly enforced?

Yes. Staff at the entrances do turn away visitors who aren't dressed appropriately, so it's worth checking before you leave your hotel rather than risking it.