I have the large LN. It is big and heavy - perfect!! I use it mainly for it's namesake task - shoulders on tenons. In this application, the plane is laying on its side on the tenon's cheek, with the blade to the shoulder [obviously]. One hand holds the plane to the shoulder, and the other hand's index + middle fingers hook in the large round-ish opening in the plane body, and pulls it across the shoulder.
I like the feel of the plane on a "pull" stroke, so I also use it that way when I need to stand it up beside the shoulder to tune up the cheek itself - but I generally don't use it on the rest of the cheek - Mr LA Block gets that job, or the router plane [and a cheek float got here the other day, so I'm going to try that out soon]. There are rare occasions where I have a long rabbet that was cut on the TS [like, for drawer bottoms]. In that case, I stand it up and pull if I can - if I can't, then my hand can fit around the brass handle good enough [pretty much like the photo in the Schwarz article] - just a couple strokes to clean up the TS blade scores.
I now use it on virtually very tenon I cut - it's set for a very thin cut and the mouth closed way down - a nice, quick, clean surface.
Credit where it is due - somewhere in the wealth of information he has available on his site, Derek Cohen has a thing on tenons - that was a eureka moment for me with the shoulder plane - and also caused an LN large router plane to arrive right away.
I have no experience with the LV shoulder plane[s], so cannot comment on them. Also - for what I use it for, I've always had a suspiscion that the medium might work a bit better than the large, but I haven't gotten far enough down the "nothing else left to buy" list to spend the $$ to find out.
When I started woodworking, I didn't know squat. I have progressed in 30 years - now I do know squat.