We believe that there are more copies of SQLite in use around the world than any other SQL database engine, and possibly all other SQL database engines combined. We cannot be certain of this since we have no way of measuring either the number of SQLite deployments nor the number of deployments of other databases. But we believe the claim is defensible.
The belief that SQLite is the most widely deployed SQL database engine stems from its use as an embedded database. Other database engines, such as MySQL, PostgreSQL, or Oracle, are typically found one to a server. And usually a single server can serve multiple users. With SQLite, on the other hand, a single user will typically have exclusive use of multiple copies of SQLite. SQLite is used on servers, but it is also used on desktop PC, and in cellphones, and PDAs, and MP3-players, and set-top boxes.
At the end of 2006, there were 100 million websites on the internet.  Let us use that number as a proxy for the number of deployed SQL database engines other than SQLite. Not every website runs an SQL database engine and not every SQL database engine runs a website. Larger websites run multiple database engines. But the vast majority of smaller websites (the long tail) share a database engine with several other websites, if they use a database engine at all. And many large SQL database installations have nothing to do with websites. So using the number of websites as a surrogate for the number of operational SQL database engines is a crude approximation, but it is the best we have so we will go with it. (Readers are encouraged to submit better estimates.)
Now let's consider where SQLite is used:
By these estimates, we see at least 500 million SQLite deployments and about 100 million deployments of other SQL database engines. These estimates are obviously very rough and may be off significantly. But there is a wide margin. So the SQLite developers think it is likely that SQLite is the most widely deployed SQL database engine in the world.