Use the plural when referring to the paper and its content. No exceptions. "Our paper", "we used", etc.
I have never seen a multi-author paper referred to with the singular. Readers will either think that it is a mistake, or that you are referring to a different solo paper; if they realize it is intentional it will come across as extremely arrogant and disrespectful to your coauthors.
Before using a plural pronoun, it is a good idea to make it clear who your coauthors are, so that the pronoun has a clear antecedent. "In a 2016 joint paper with A. Smith and B. Jones, we showed that..."
You may certainly point out that you were the first author. "In a 2016 joint paper with A. Smith and B. Jones, of which I was first author, we showed that..." But you may not write as though your coauthors don't exist. They must have made significant contributions to the paper; if not, they shouldn't be coauthors at all.
If you have space in your letter or other application materials, then after you summarize the content of the paper (using the plural), it is fine to describe in more detail your own specific contributions using the singular. "I designed the experiment, analyzed the data, and wrote the majority of the text of the paper." Note carefully the (sometimes subtle) distinction between the results and achievements of the project as described in the paper, which are collective and take the plural ("we showed that X", "we performed a placebo-controlled trial of drug Y", etc), versus the specific work you did in order to help accomplish those things, which illustrate your personal skills and take the singular ("I analyzed the data").
If you think it is starting to sound like you did everything while your coauthors did nothing, you can clarify briefly: "...while my collaborators worked mainly on gathering the data and developing the statistical methods, and wrote the corresponding sections of the text".
However, the very best way to "claim" your contributions is to have a recommendation letter from one (or more) of your coauthors, in which they explain how the paper was primarily your work and they just contributed bits and pieces, and then go on to praise your capability as an independent researcher. This saves you from having to brag too much, and serves as verification that you really did all this work and aren't just lying.