Does the Laplace transform biject? Someone wrote on the Wikipedia article for the Laplace trasform that 'this transformation is essentially bijective for the majority of practical uses.'
Can someone provide a proof or counterexample that shows that the Laplace transform is not bijective over the domain of functions from $\mathbb{R}^+$ to $\mathbb{R}$?

It depends on you set of definition. If you ask wheter it is bijective from the functions defined on $\mathbb{R}$, compare the laplace transform of 
$$
f(x)=\begin{cases}
\sin(x)&x\geq0\\
0&x<0
\end{cases}
$$
and
$$
g(x)=\sin(x), \forall x.
$$
