Back in Spotify’s early days, when the company was just a dozen people in a small office in Stockholm, Daniel Ek, a co-founder, liked to compare it to Apple and Google.

It was 2008, and the traditional music industry was collapsing. Yet as Spotify introduced its streaming service in a handful of European countries, it clung to what must have seemed an impossible ambition: challenging the titans of Silicon Valley to become the world’s leading outlet for online music, with a hybrid free-and-paid model that made record companies nervous.

After a decade, the start-up from Sweden has proved itself a worthy adversary, with 157 million users around the world, 71 million of whom pay for subscriptions. That is about twice the number of its closest competitor, Apple, which finally entered the subscription game three years ago.

And on Tuesday, in a ritual of success for any start-up, Spotify’s shares will begin trading on the New York Stock Exchange with a valuation that could exceed $20 billion. Underscoring the company’s self-image as a disrupter, it has shunned the usual circus of an initial public offering in favor of a rarely used — and potentially risky — process known as a direct listing, in which no new stock is issued and insiders can begin selling their stash on Day 1.