You will probably only be able to use material under CC BY ("Creative Commons Attribution") licenses and no other CC licenses. Any other Creative Commons terms will be problematic:
Non-commercial (NC) terms eliminate your ability to distribute the icons commercially, which is likely a problem for your proprietary business model, as well as being (A)GPL-incompatible.
No derivatives (ND) terms prohibit the creation of derivatives, which probably disallows the inclusion of the work within your software as a larger work.
ShareAlike (SA) terms require that derivatives be distributed under the same CC ShareAlike license (or any license the Creative Commons foundation enumerates as a "compatible license"). This rules out your proprietary license, and sort-of rules out the AGPL as well (the workaround is that the GPLv3 is a compatible license for SA derivatives, and the AGPLv3 permits you to include GPLv3-licensed material in a larger AGPL work).
Depending on how you use the CC-licensed assets, there is a possible legal argument to be made that the icons and code are separate works that happen to be distributed side-by-side, and the code merely uses the icon files as input data. However, I don't know that this has ever been tested in court, and it may vary based on exactly how you use the assets. Perhaps packing them in a single executable file may yield a different legal result from distributing them as separate files in a directory. I am not a lawyer; I don't know and cannot make a strong speculation.
If that turns out to be true in your case, then you could use icons under CC licenses with ND and SA terms, since their requirements about derivative works would not apply to your software insofar as your software is not a derivative work. NC terms would still be problematic from a business perspective, since you could not include the icons in a for-pay distribution, regardless of the software's status as a derivative.
In any case, I am fairly sure that a CC BY license should be fine, since it limits the scope of its requirements to the original work, not derivatives. The FSF lists CC BY 4.0 as GPL-compatible.