First, go to the webinterface of your phone.
Choose Software Update in the navigation on the left.

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Next, you have to find the current firmware version for your model in the snom wiki. To do this, you have to follow the ? next to the Firmware text field to this page Show archive.org snapshot and then follow the link of the current firmware below Firmware Versions in the wiki. Now select your phone model. Next you have to choose between the minor versions. Follow the link of the most recent stable version (or any other if you feel like experimenting). Now you will see a page with instructions (e.g. snom 300 Firmware 8.7.5.35 (2017/07/11) Show archive.org snapshot ). Now you have to copy the link for your model.
Then go back to Software Update in your phone settings web interface. Paste the link to the firmware text field (it looks as if it was disabled but it isn't). Click on Load and wait until your phone has finished updating. If there were further instructions on the wiki page, you should consider taking them.

That's it.