Statement of Purpose, should one say "I did X ..." or "My advisor and I did X .. "?

This is an XY problem. Saying "I did X" repeatedly is wrong and offputting, and saying "My advisor and I did X" repeatedly sounds weak and annoying to read. Neither is the right option alone.

You should adjust the phrasing so that subjects are simple, mostly either "I" or "we", and explicitly credit yourself for things you actually did on their own. After all, you did have some independence, right? It's not like your advisor was over your shoulder every minute of every day.

Bad example:

My advisor and I were looking for a project in field X. My advisor gave me a project to apply X to Y, and my advisor and I planned the general strategy after my advisor taught me the basics, so that after two months, some progress was made, though my advisor decided that the data analysis would have to be redone. Later, a paper was drafted with the editorial support of my advisor, which is now, after having been submitted, under review, at Journal Y.

Better example:

I wanted research experience in field X and approached Prof P, who agreed to be my advisor. My project was to apply X to Y, and after a brief reading period I began two months of independent work, during which I picked up techniques A, B, and C. However, I found these techniques were not powerful enough for the problem at hand. We decided on a change of strategy, and I modified the analysis with techniques D and E, which I used to produce compelling results. We have written up and submitted these results to Journal Y.

Of course you would use fancier language in the real thing, but note how the second example makes you sound like an actual sentient human being, while the first makes you sound like you were just passively "along for the ride", even though it technically makes the exact same claims.