Here are various reasons that I may choose to use a hand strap, shoulder strap (never around the neck), monopod, or tripod:
Hand Strap - I don't currently have a hand strap and have not used one for a couple of years, my gear is just too heavy at this point. Fast lenses on prosumer DSLRs are too heavy for me to carry with just a hand strap for 5 hours at a time. But if I had a lightweight system, like a 4/3 or a rangefinder style camera I would exclusively use a hand strap for the ease of use and quick access.
Monopod - I shoot youth sports and have had many days when I think very hard about dragging a monopod around, but when I look at a photographer who is using a monopod at a game it always seems to be slowing them down and keeping them glued to a single location. I prefer to stay in motion and cover the entire game, a monopod just gets in the way of that. If I was going to be sitting in one location, or was stuck in a press area, I would totally use a monopod, but otherwise not.
Tripod - I have 4 different tripods, one for all occasions. I use them for low light landscape and architectural photography. I have not used a tripod for any other reason in the last 5 years and don't expect I ever will again.
Shoulder Strap - my main means of carrying my camera gear and camera around are with shoulder straps. On my bags I use Optech or similar neoprene straps that have a lot of give to them. They are very comfortable and the stretch makes a huge difference when carrying heavy loads. I wear them bandolier style across my chest.
For carrying my camera I use an Up Strap LARGE PAD CAMERA STRAP + HEAVY DUTY RELEASE + KEVLAR ENDS. I tried a lot of different straps before this one... OEM straps are harsh and chafe my skin, neoprene straps won't stay on my shoulder, many straps do not have a quick release that I require for low light or when I just want the strap out of the way. The Up Strap has solved all of these issues. It stays on my shoulder like glue, the rubber pad does not chafe my skin or hold moisture. Even with a 100-400 lens on a 5d Mk III with a battery grip I feel very secure with this strap and I can carry it all day long.
I recommend giving different straps a try, if you are interested in a monopod, give that a try. But don't cheap out, don't buy some piece of junk that costs half of what others cost and expect to have a good experience. When you are out taking pictures, ask other photographers how they like the gear they are using. Experiment.