Even for a smooth affine curve $C=Spec(A)$ over a field $k$, you can get nontrivial elements of $K_1$ (where nontrivial means not coming from $A^*$), all of which are detectable by Mennicke symbols.  For example if $k={\mathbb R}$ and $C$ is the affine circle $x^2+y^2=1$, then the symbol $[x,y]$ is nontrivial.  You'll find this example, and much else on the $K$-theory of curves, in Bass/Milnor/Serre's paper on the congruence subgroup problem.