This page in a nutshell: Knowing how to "lose" a debate, with civility and grace, is sometimes as important as "winning" it.
Even if you are "losing", it is important to learn how to lose with grace and dignity.
Every Wikipedian needs to know how to lose with a semblance of grace and dignity.
We like to think of the Wikipedia community as being infallible, but in the short term, mistakes are often made: A tangent derails a discussion for a much-needed clarification of policy; the community fails to show up and defend itself against poorly conceived plans; an editor's strong reputation or an early knee-jerk reaction prevents a clear analysis of a proposal; an advice page recommends an action well-suited to one problem, while inadvertently creating serious problems in other situations; or emotions run high during editing of a controversial article, and an edit war breaks out.
When you are on the losing end of an argument, remember these things:
Let it go—for now, at least. So what if your ideal improvement can't be made today? If your idea is a good one, it will still be a good idea next year.
Failure isn't the end of the world. Failure is unpleasant, but there are still 5,972,073 articles out there, and 99% of them need to be improved.
Stick to real, current, practical problems. There might be an ideal solution to a hypothetical problem, but sometimes, nobody may be interested. In discussions, give simple, concrete examples.
TL;DR is the law of the internet. If a discussion gets too long, everyone loses. Keep your responses brief, and stick to the main point. Avoid the "chunk of text" defense.
Silently limit yourself to one or two comments a day on a particular page. You've got to give other editors a chance to participate and to defend you and your idea. Work on one of those 5,972,073 articles instead of endlessly refreshing the page so you can pounce on any reply. By doing this, you force your opponents to slow down, too, and perhaps they'll use the extra time to think more deeply about the issue.
If you're feeling stressed by the dispute, take a break. Wikipedia is not that important, and it will be here tomorrow. Who cares what the page says for the next 24 hours? Plan something fun to do, catch up on your real life, and come back tomorrow, or next week.
A variant of the above approach is to pick an article in an entirely different section or part of the project to work on. If disputes over a music article are stressing you out, try editing a history article or a geographic article; after all, on the English Wikipedia, there are almost five million articles to work on. Alternatively, try editing a different part of the project, such as user essays. Or find a different project to work on. The article on your hometown or favorite vacation spot probably needs some help at Wikivoyage. Wikisource's Wikisource:Proofread of the Month group is friendly and helpful to new folks. Add photos to articles at non-English Wikipedias. Give it a try.
Copy the article that is in dispute and paste it into your Sandbox on your user page. Then you can make all the text and source changes that others are against you making, and make the article "perfect," at least from your point of view.