By its very nature, this question cannot expect a definitive answer but here are some suggestions.
For a curve with parametrisation of the form $(\int^t(u)du,f(t))$ for a function $f$ of one variable it is the case if $f$ is injective or, better, if and only if $\int^s f(u)du\neq \int^t f(u)du$ whenever $f(s)=f(t)$.
Many important curves have parametrisations of this form (circle, cycloid, catenary, ....).
In a certain sense "every" curve has such a parametrisation. More precisely, consider the curve with parametrisation $(x(s),y(s))$. Since self-intersection is preserved under diffeomorphisns, we can suppose that the curve lies in the upper half plane. Under the new parameter $t$ where the latter is, as a function of $s$, the primitive of $\frac{x'(s)}{y(s)}$, the parametrisation will have the above form.
Of course, this will not work universally since this $t$ will not always be a reparametrisation but it will be in many concrete situations, e.g., if $x$ is strictly monotone.