Would it be better to use 1H NMR or 13C NMR spectroscopy to distinguish among 1-butene, cis-2-butene, and 2-methylpropene

13C NMR

Explanation:

Both types of NMR will do the job, but 13C NMR makes the clearest distinction among your compounds, shown below as A, B, and C.

13C NMR

13C NMR is unambiguous, because we can distinguish by the number of peaks in the spectra.

Compound A has four non-equivalent carbon atoms. It will give four peaks.

Compound B has two sets of equivalent carbons: the methyl groups and the alkene carbons. It will give two peaks.

Compound C has equivalent methyl groups. It will give three peaks.

1H NMR

Compound A will show four groups of protons in a 3:2:1:2 ratio with complicated splitting patterns

Compounds B and C have equivalent methyl groups and alkene hydrogens. They will each have two peaks in a 3:1 ratio. However, the methyl groups in B will be a doublet (J = 7 Hz) while those in C will be essentially a singlet (J ≈ 1 Hz).