Look for the phrase "scale-up" instead of "start-up", these are usually well-funded, up-and-running companies that still have some of the start-up mentality but without the financial worries.

Of course, some of these are already a decade old and may employ hundreds or thousands of people already, so YMMV; they might just be like older, boring enterprises, just not as formal or fixed in their ways.

Look at their investors. When interviewing, ask to speak to the investors. See what they say. When speaking with founders do they have numbers they can show or is it pure show-personship?

I mentioned this above, but unlike an investor you don't have to pick a start-up that exits successfully for you to gain really valuable experience that can help you on your own.

Pick a company with a team you respect, a product you connect with, and has money in the bank (just completed a funding round).

You can't go much wrong if you do that, at least in terms of learning.

Picking a product that you personally use, love, and would pay for helps a lot