I have a public profile on Instagram and I know about the option to make all your photos private on the service. However, I don't want to make all my photos private, but selectively make some available to the public (the rest as private). Is it possible to do so?
No, as of now it is not possible. As of now you can only have control over the followers who view your photos, but not the individual photos. Setting your profile to private will prevent unwanted viewers from seeing your entire profile and photos on it, but if you choose to share an Instagram image to Facebook or Twitter, the URL for that image would then become public, essentially allowing you to choose which images are public/private outside of the Instagram network.
See more details on Instagram Help Center.
You can now "Archive" an Instagram post.
The picture will no longer appear on your profile grid. They are private to yourself.
Only you can see the posts you've archived.
Press the "..." button on a post to have the option to Archive it.