The most popular way to transfer files over the internet?

Answer 1

In pure popularity terms this would have to be either HTTP or FTP. These are by far the most widely used protocols for transfer of files.

However suitability of a protocol depends entirely on the application, for example it would be better to use a protocol such as RSYNC to transfer files between servers given that this can additionally use an encrypted tunnel to send data.

Suitability also depends on the size of the files involved, for instance small files could quite easily be sent via e-mail. Whereas large files would be better sent using a protocol that supports error checking.

In short, it depends on the application.