My first single shot was a sharps. I loved it, and managed to do quite well with it in Schuetzen and hunter's silhouette shooting. However, I eventually realized why they say the extractor and firing pin of the Sharps is its biggest weakness. I have replaced quite a few 1874 Sharps rifle firing pins and extractors over the years. I hardly ever shoot mine any more, in favor of the High wall, trapdoor and Mauser.
The next single shot I started shooting was the trapdoor, which in my opinion is the most highly under-rated rifle ever. It is completely reliable, has the best ejector of any rifle I know of, can be fitted with a tang sight and set triggers, (the original barrel sights are pretty good and some have windage adjustment), parts are readily available and interchangeable,( if you know the difference between cadet parts  full size parts, two and three stage tumblers and the two widths of receivers). I have been shooting one of my trapdoors in High Power for several years and you can fire ten well aimed shots in a minute with a little practice. It loads and unloads as fast or faster than any single shot I know of. Its primary weakness is the inability to mount a scope over the bore. As far as strength is concerned, its plenty strong for anything I am going to reload. 
As for Schuetzen rifles, I would not trade my Uberti High Wall for anything. Also have an original .22 High wall Musket and a Uberti replica low wall. Never had any problems with them, parts are available if needed, which is rare, and they are strong and accurate. 
As for other rifles, you cant beat the 98 Mauser. It is strong, reliable, accurate and parts are plentiful. Been shooting mine in high power with cast bullets for years and had a lot of fun with them. Almost every country in the world has used them as their main battle rifle at one time or another, and for good reason. 
These are my favorites , your mileage may vary. 

Joe S

My favorite is the Stevens 44 1/2.