I've never had any awkward or uncomfortable situations standing in lines alone, except for those situations that would occur no matter how many people were in line with me (like kids behind you who continually move up right on top of you, or people with huge backpacks who swing them around like weapons, or deliberate line cutters, or odoriforous people with less-than-stellar personal hygene).
Sometimes I strike up conversations with people in lines, other times I just wait patiently and look at my digital pics or drink from my water bottle. Once in a while I even make trip notes, though I usually prefer to sit down someplace for that.
Standing in lines alone is no different than doing anything else alone. Standing in lines is not an inherently "group" activity, any more than waiting for your food at McDonalds or waiting to check out at the grocery store.
What you are really nervous about is not "standing in lines alone", it is simply "being alone", which aflicts many people, particularly those with families. Some focus on a particular aspect of a solo WDW trip and think it's the "worst" part (travelling alone, checking into a hotel alone, staying in a hotel alone, touring the parks alone, etc.) But it's not really that aspect that's making you nervous, it's the basic concept of being alone.
Once you get past any fear or nervousness you might have about doing things on your own, without spouse, kids, parents, or friends to slow you down, then you can really relax into a solo WDW trip and start to enjoy the wonderful differences.
Standing in line for Splash Mountain alone is no different than standing in line alone at the post office. You're just in a more pleasant surrounding, and have a more pleasant reward at the end of the line.
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