The subject here is punctuation, not spelling.
Joe’s Bar is a bar belonging to (or founded by or run by) one guy named Joe. Joes’ Bar is a bar belonging to two or more guys named Joe. Joes Bar is a bar belonging to an idiot.
Many people type “Mountaineers Route” out of ignorance. If it were named for John Muir’s first ascent, I would expect it to be “Muir’s Route” or “The John Muir Route” or something similar. I seriously doubt that the name refers to Muir. I always type “Mountaineers’ Route,” since it belongs to and is climbed by all of us.
Bob West is correct about where it starts.
