How can I see a log of emails that Drupal 7 has sent? It would be handy for things like troubleshooting registration issues... to see that registration emails are actually being sent.
Drupal doesn't keep hold of that information, so your server mail logs are the place to look.
Some email operations are recorded to the watchdog, but not all. You can get a look at a few of those ops with a simple query like this
SELECT * FROM watchdog WHERE message LIKE '%email%'
That'll get you false positives, but also should show you what operations are actually recorded (welcome message sent, new password sent, etc.).
For future reference, you can use either of these modules to log outgoing mail:
There is also the mail_logger module