If you could do a frequency sweep on a circuit, I could understand why laplace bode plots might be useful; you may get a plant transfer function or something.
A single laplace transform doesn't seem too helpful, eg. a pure sinusoid: sin(wt) -> w/(s^2 + w^2). Do you just want that equation on the scope? :S
The key distinction between laplace and fourier frequency domains is that the former tends to be considered unilaterally (ie. 0 to infinity), while the latter tends to be considered over all time (-inf to +inf). With laplace, the substitution s=jw can be made, as amspire said (where j=i and w=2*pi*f).
Certainly, laplace transforms are great for solving initial value problems, but on a scope, I can't think of a situation where that would be required – you have access to experimental data in any case. The real benefit of the FFT is the decomposition of the signal into fundamental frequencies, which has many uses, such as determining how pure a waveform is, identifying sources of noise, etc.