I agree with MikeD. A chain made from half links just adds weight & a fashion statement. If a fashion statement is what you want there is no trouble with that, but there is not technical advantage.
First, half links are made from bent plates rather than flat as in a normal pitch chain. So they either stretch quicker (the pedalling force wants to straighten the plates) or are very heavy (over engineered plate thickness to counter act the straightening force). Usually they are just the heavy type. A light(er) one that stretches quicker will of corse wear the rings quicker & lead to jumping quicker. Most likely in turn leading to painful experience. So if pain & fashion are your thing use a half link chain for proper MTBing. If you just want to pootle down the pub or around town will be no difference in performance or wear really.
Secondly, 1/8" or 3/32"? I've used both & really can't tell the difference. There is a difference technically I'm told that 3/32" is stronger as it is designed to work when dragged across cogs under load when not in perfect alignment. So this suits most DIY single speed set ups as chain alignment is quite often not perfect. Where as a 1/8" chain, particular a good quality track version is not very forgiving. I have not real idea why track bikes or BMX bikes ect use a 1/8"chain & geared bikes use 3/32" chain but can imagine that the geared chain was an evolution of 1/8" chains. Can you imagine how dished a rear wheel would be if it needed to run a 7/8/9sp cog based on a 1/8" chain width? Anyway I think any minute advantage in getting the power down on a single speed through a 1/8" chain over a 3/322 chain is completely lost as your riding a mountain bike in the mud. I use 3/32 chains with no problems.
So there you go, my complete and unadulterated opinions on chains for a single speed. Hope you found it helpful. I like having a spray every once & a while.
P.S. what every you do use rings with decent teeth - now that’s better advice than any of the stuff I just bleated on about above.