I have a single VPS running a number of independent Drupal sites. What considerations do I need to make (if any) when installing APC?
One thing to keep in mind is that APC will require quite a bit of memory if you have multiple sites because each file will be kept in memory, even if it is in fact exactly the same file, just with a different path.
So you need to watch and tune the amount of memory that apc uses/can use. By default, it's 32MB, that's often too less for even a single Drupal site if you have many contrib modules. The problem is that if APC has not enough memory, it will always have to delete files from the cache. If it has much to less (e.g. not even enough for all files in a single page request), it will make your site slower than if you hadn't used APC at all.
VPS often don't have a lot of memory, make sure that you have enough for your sites. You could consider merging them into multi-sites to save memory.
The implications are a much faster website! Check out this article, and others, by 2bits.com
Here is another that talks specifically about tuning APC for drupal sites
My company runs several sites on a single VPS with APC configured as suggested in the above articles. That, with memcache, significanly speeds up our sites.
We have not witnessed any server issues or conflicts or had any security concerns, if that's more what you are asking.
APC as a general opcode cache is entirely safe, and will greatly help the performance of your sites. However, there is one potential problem. You can't use Drupal modules like http://drupal.org/project/apc in most shared server APC environments as it's a big security risk. The APC cache is generally shared between all sites running on the server without a lot of thought going into separating them. If you are not a sysadmin, I would steer well clear of these modules.
That said, just installing APC for the opcode cache will generally increase the performance of your sites by quite a bit, so it's definitely worth using. Just be aware of some of the issues with the other functionality provided by APC.