APC and the trunk (development) version of WP-Super-Cache run together here without errors on multiple Linux (Ubuntu) and Windows servers. WP-Super-Cache 1.0 fails on all those servers with or without APC, but the problems that cause those failures have been fixed in the trunk version. (Anyone using APC should be sure to use the latest version, because earlier versions had a serious bugs. Some of those bugs cause unrecoverable malfunctions that necessitate server reboots.)
As you probably know, there are many APC configuration options. APC will fail for a variety of reasons if those options are not set correctly for particular installations. Some settings cause server failures that seem random and also misleadingly occur only when certain PHP scripts, such as WP-Super-Cache scripts, run under certain conditions of available memory and other things. APC works reliably and significantly speeds PHP processing when it is configured right, but finding the best settings for particular installations can be challenging.
The best way to get APC, Apache, PHP, MySQL and everything else configured to run optimally is to install them on an identical test server that has WordPress/WP-Super-Cache installations mirror-imaged from the normal server and then to use stress-test software that simulates high levels of website traffic to measure speed performance and capacity limits, find causes of failures, and experimentally improve results. That is obviously a lot of work, but the results can be very rewarding, because it is common to find simple configuration changes can greatly increase speed, capacity and reliability.
Many configuration options in Apache, PHP, APC, MySQL, WordPress, WP-Super-Cache, and other things are complexly interdependent. It is impossible for most people to make reasonable guesses as to what is even approximately best. The interrelationships are so complex that even someone with intimate technical knowledge of the inner-workings of all the components is apt to make selections that testing will prove are not optimal.
The difference between a server that will crash with a few users and one that will quickly serve thousands without difficulty is often a few configuration settings. Stress-testing is the only way to know the truth about speed-performance and capacity limits. It is lots of work, but where an optimally-configured old desktop computer with WP-Super-Cache can easily outperform a new mega-buck server that is not optimally-configured, stress-testing to find optimal settings is worth the effort.