If you updated your Verizon Galaxy S4 to the latest over-the-air build, or bought one after it was already installed, you might already be aware that it's been especially resistant to root attempts. Savvy power users have stayed well away from the updated VRUAME7 firmware, which makes rolling back to an earlier build impossible. But as is so often the case, two enterprising XDA-Developers posters have solved at least some of the problems, and released a new rooting method for the latest OTA.
The exploit comes from (or at least, was posted by) XDA members DoomLoRD and PWM978, among others. It's a bit unconventional: you'll need to use a custom-formatted MicroSD card with the root exploit files loaded up, plus a terminal emulator on the phone itself. The process isn't all that difficult if you're used to the more typical forms of rooting (as opposed to one-click solutions and the like) but it's clear that this technique took a lot of work. You'll want to follow the steps in the XDA post to the letter.
While this will restore access to root apps for those running the latest Verizon firmware, these users are still out of luck when it comes to an unlocked bootloader, and the wider custom ROM capabilities that come with it. At the moment there's also no custom recovery option. Locked bootloaders are notoriously difficult to work around, and while developers have become better at cracking them, it's certainly possible that no solution will be found. There's currently a $3,915 community bounty being offered for a full modding solution.
As always with this sort of thing, it's best not to proceed if you want to hang on to your warranty. If you're ready to take the plunge, hit up the source link below for the necessary files and instructions.