What are the main differences between BIGIP versions 15, 16, and 17 ? What is the current recommended and stable version to upgrade to?
Usually, the most stable version in each software release is the latest one which contains all of the fixes on that major software version.
For example, here is the list of the most stable versions in each major release, which are the latest releases:
version 15: 15.1.7
version 16: 16.1.3
for version 17, it is the base version only that is released.
Regarding the differencies between major versions with respect to the features, you can check the release notes for each version you are looking forward to upgrade to:
Or if you are looking for an exact feature, you can simply check in which version it is published and then select this version.