Backing color? Whatever I find on sale. Have white, olive, rainbow and blaze orange.
Fly lines I have range from a solid muted olive to a full blaze orange. Some of the more offensive ones: SA trout in solid / bright yellow, Airflo bass which is orange and baby blue, Airflo river and stream which is seafoam. Bam mentioned non two-toned lines but I believe that selection is pretty small nowadays. When casting at distance and having to guess where the fly is, bright colored line seems to help me ballpark the fly location a little better. Haven't really noticed any significant changes in success because of line color.
The fish on Clarks Creek always had a tendency to be extremely spooky. I gave up on worrying about clothing color after dressing in full camouflage and still having them spooked just because of the movement of the cast.
I'm just old school & stubborn... I started with white Micron backing and it just seems to be what I always buy...
It's also a "virginal" thing... I figure I'll never need to consummate my relationship with my backing while fishing so I assume it should be white.
In my perfect world all fly lines would be white or cream colored because I can see them and the fish never seem to mind. I own a few, but they are tough to find in the "true-to-weight" DT's I prefer few so I look for lighter colors like pale yellow and I like the "willow" color SciAnglers offers. I also fish real silk lines and even though I can get them dyed any color I want, I always buy the natural yellow color.
I also have and still use a few of the long discontinued Rio Lumilux lines that are a pearl white color and glow in the dark. They are still my hands down favorite plastic line and even in total darkness during a hatch, that "glowing" line doesn't bother the fish...
The two toned lines I don't like & won't buy are DT's. I do reverse my DT's and prefer they are the same on both ends and that includes the taper. I do use a WF Rio Smallmouth Line that is two toned and in this application I like the two colors.
In regards to clothing colors, for a LONG time I fished everywhere INCLUDING the Letort with a bright white hat, a light sage fishing shirt and a red bandanna around my neck, along with a white or yellow fly lines and never felt impaired. These days I have a different hat, but nothing else clothing-wise has changed.
I've also fished Clarks a few times with a blaze orange cap & bandanna around my neck after encountering some hunters by the stream. When hitting wild trout streams during different hunting seasons I'll add a a blaze orange vest to the mix and I never felt at a disadvantage so I never worry about clothing color anywhere.