Saving settings in user.config
First, I think making it a singleton would be a mistake (singleton usually is). At best it would be extra work with zero benefit. At worst, it's a handicap, such as when/if you decide to support two top-level windows with independent settings for each.

Second, I think I'd add a Boolean named dirty (or something similar). When you set the value, dirty is set to true. When you save the value, you check if dirty is true, and only write out the value if it is (then you set dirty to false).

Personally, I'd probably change it from a static class named ProcessPath to a normal class named (something like) config_string, with an instance for ProcessPath, and a the possibility for other instances holding other strings. To do that, you'd pass Settings.Default.ProcessPath as a parameter when you construct the object.