For the past two minor release Drupal core upgrades, I've had major problems trying to get some of my Composer-based Drupal codebases upgraded. For both 8.3.x to 8.4.0, and now 8.4.x to 8.5.0, I've had the following issue:

1. I have the version constraint for drupal/core set to ~8.0 or ~8.4 in my composer.json.
2. I run composer update drupal/core --with-dependencies (as recommended in Drupal.org's Composer documentation).
3. Composer does its thing.
4. A few things get updated... but not drupal/core. It remains stubbornly on the previous minor release.

Looking around the web, it seems this is a very common problem, and a lot of people soon go for the nuclear (or thermonuclear1) option: