From J.D. Salinger's The Catcher in the Rye, what are a few examples of how Holden Caulfield avoids forming relationships with others?

Holden suffers from depression and social anxiety. These two mental illnesses certainly don't help Holden to sustain relationships. In fact, Holden either avoids deepening a relationship with new people by running away or he sabotages his current ones by acting out in some immature way. Looking more closely at Holden, one might discover a pattern that he tends to follow. The destruction of the relationship generally starts in his head as he criticizes the other person for behaviors that he doesn't approve of.  Then, he loses control verbally by insulting the person, physically by hitting the person, or simply by avoiding the person all together.

One relationship that he hurts through verbal irresponsibility is with Sally. Holden says the following:

"'C'mon, let's get outa here,' I said. 'You give me a royal pain in the ass, if you want to know the truth.' Boy, did she hit the ceiling when I said that. I know I shouldn't've said it, and I probably wouldn't've ordinarily, but she was depressing the hell out of me. Usually I never say crude things like that to girls. Boy, did she hit the ceiling" (133-134).

The date breaks up after that and he probably never sees Sally after that. An example of how Holden ruins a friendship by hitting someone is with his roommate Ward Stradlater. Holden gets jealous because Stradlater has a date with his friend Jane and he doesn't feel he will respect her. After the date, Stradlater won't give Holden details and Holden flips out. 

"This next part I don't remember so hot. All I know is I got up from the bed, like I was going down to the can or something, and then I tried to sock him, with all my might, right smack in the toothbrush, so it would split his goddam throat open. Only, I missed. . . Anyway, the next thing I knew, I was on the goddam floor and he was sitting on my chest, with his face all red" (43).

In the end, Holden feels guilt and regret for his behavior, but he doesn't do anything really to mend these two relationships. 

The final relationship which is ruined by Holden running away is with Mr. Antolini; however, this is one where Holden might have done the right thing even though it put himself in danger. Holden wakes up to Mr. Antolini petting his head and he freaks out and runs away in the middle of the night. He jets out of there quickly instead of standing up for himself and holding Mr. Antolini responsible for his actions (192). On the other hand, Holden is the child in this situation and needed to run, but it is another example of how Holden compulsively runs away from people.

For other relationship examples, see what Holden says about Jane Gallagher, the girl he never calls (78-79), a schoolmate, Carl Luce (140-141), and his siblings (198-200).