Will it improve my PhD applications if I take a gap year to work on research? I was a philosophy major who took a bunch of graduate-level classes in linguistics and I wanted to apply to PhD programs in linguistics (in the United States). I had a writing sample in the philosophy of language, and I found out later from other professors that PhD programs in linguistics are mostly only interested in BAs and MAs in linguistics (unless they have a very strong linguistics writing sample), regardless of what courses they took. So I ended up getting waitlisted to a couple of schools (one top-ranked) and rejected everywhere else, and turned down at both places I got waitlisted. I informally applied to a bunch of MAs in linguistics in May, just by sending emails, and I got accepted to many but without any kind of funding, so I wasn't able to go anywhere this year.

I have no plans or anything whatsoever this year, and to be honest, it's depressing and boring. I'm also unable to get a job.  I'm doing my best to do research and work on two new writing samples for linguistics, which I hope will make me seem like a more serious applicant.

I was just wondering: will this situation look bad on my application? Should I describe what I went through in my statement of purpose? I want to reapply to a school I got rejected from and another school I got waitlisted from. Would that be all right if I really improved my application?
Your situation sounds similar to mine. After finishing my undergrad and not getting a PhD position, and since it was too late to apply for a full MSc, I moved to another country, enrolled just in the thesis and I worked on it for a year. After that, I got a lot of experience, a good recommendation, and a position. Now, I was lucky I could move and found a university that the only requirement for registering was finding a willing supervisor. So, I won't say necessarily that everything is lost, but it may be difficult depending on your situation.

I would advise against being fully independent researcher. You are fresh from undergrad, and with not so much experience in the field. If you work without adequate supervision chances are you will be reinventing the wheel, walking down paths that lead nowhere, or creating a beautiful yet useless theory. Your priority should be finding a professor that can guide you and provide some interesting problems to work on and feedback.

You should also keep an eye to applications in Europe: positions are opening and rolling all year round.