I would like the equation of a non-isotrivial elliptic curve over the rational function field $\mathbb{F}_2(t)$ with exactly one place of supersingular reduction and I would like to know which place.
I tried $$ Y^2 + tY = X^3 + tX + (t+1) \, . $$ I think it is supersingular at the place $t+1$. Because $t \equiv 1 \bmod {t+1}$ and so the curve reduces at $t+1$ to a curve of equation $Y^2+Y=X^3+X$ which is known to be supersingular. However this curve is isotrivial: its $j$-invariant is zero. I would like a non-isotrivial elliptic curve.
Consider the elliptic curve $E: y^2 + t xy + y = x^3 $. Its $j$-invariant is $\frac{t^{12}}{t^3+1}$, so it is not isotrivial. Since the only supersingular elliptic curve over $\overline{\mathbb{F}_2}$ is $y^2 + y = x^3$ with $j$-invariant $0$ (cf., Exercise 5.7 in Silverman's Arithmetic of Elliptic Curves), this shows that the only place of supersingular reduction is $t=0$.