I've seen expressions like $x\lt y\in S$ used on this site to mean "$x\lt y$ and $x\in S$ and $y\in S$." This has me worried because I sometimes use $x\lt y\in S$ to abbreviate "$x\lt y$ and $y\in S$". How widespread is the former usage? Should I avoid using abbreviations such as $x\lt y\in S$ and $x\ne y\in S$ etc. because I'm likely to be misunderstood?
Yes, ambiguous notation should be avoided in places with a general audience, like journals and here on MathOverflow. It's useful to have shorthand in specific places where all readers can agree on the meaning but it is evidently incorrect to assume that everyone everywhere agrees on that same meaning.