How do I update core to only include security patches? I have a old Drupal 7 installation and I would like to update my Drupal 7 to the latest release, but I know for a fact that some of my custom module will break on the latest update.

All I want installed is the security patches up to the latest Drupal 7. How can this be done or is it even possible without much effort? Also where and how can I view what is being patched during an update of Drupal 7 core?
If you use Drush you can update all security issues with your site via (this is Core and 3rd party modules):
drush up --security-only

if you want to see what it will do, but not do any of the actual changes you can issue:
drush up --security-only --simulate

If you want an easy way to update Drupal Core to the newest release you can use a patch site such as:
http://fuerstnet.de/en/drupal-upgrade-easier