If not, in what circumstances would it not appear?
The answer can be divided into two parts: Timeline and Non-Timeline. Let's start with Non-Timeline first [that is, you don't have Facebook Timeline profile yet]
Your comment will appear on your wall regardless of the photo being public or private.
However, if it's a public photo, all friends will see your comment on the photo. If it's private, only you will be able to see it or only mutual friends of whose photo you commented on, will be able to see the published comment activity on your wall.
Your comments don't appear on the wall, regardless of it being private or public. But it can pop-up in the news feed and friends can see it depending on the privacy settings as outlined above.
However, you can see all your comments [and other activities] by clicking the
Activity Log button.