The simplest way to safely trim your fingernails without nail clippers is to use your eye teeth, fingers, and a rough surface. While it sounds odd, this is how I have trimmed my finger nails for most of my life. I do not do this for my toe nails.
Please note that I leave a 2mm white section even when my nails are trimmed and trim them when then are around 5mm in length.
I use my eye teeth (sorry canines) to pierce the nail on the side closest to the thumb. I then use the thumb and index finger of the other hand to tear the nail off. Due to the way nails grow, the tear naturally follows the same curved line parallel to the pink part.
The resulting "cut" can have two issues: it can show a thinner paper like remnant or the "cut" is slightly rough (but has none of the sharp points I get when I use nail clippers on my smaller toes). Both of these wear off with normal use. In grade school, I used to smooth them by drawing them against the edge of the thumb nail on the same hand. In college, I used a granite rock the same way. Now I use a nail file (doesn't feel good but works better).
I don't know how well this will work for other people. My nails are abnormally healthy, even, and thick and I have been doing this for a very long time. I do this in private (or more likely alone in my car) and don't tell people. Nevertheless, the end result have been complimented many times. I've been accused of getting manicures.