  We investigate dynamical states of grand-design spiral arms in three local
galaxies: M51, NGC3627 and NGC628. Based on linear perturbation analysis
considering multiple components in the galaxies, we compute instability
parameters of the spiral arms using their observational data and argue whether
the arms will fragment by their self-gravity. Our analysis utilises
observations of carbon monoxide (CO), 21-centimetre line emission and
multi-band photometric images for molecular gas, atomic gas and stellar
components in the arms, respectively. We find that the grand-design arms of
these galaxies indicate marginally stable states, and hence they are not on the
way to fragment. We consider this to be consistent with the commonness of
spiral galaxies and the relative rarity of fragmented discs at low redshifts.
In the analysis, molecular gas is the dominant component to determine the
(in)stability of the arms, whereas atomic gas and stars are far less important.
Therefore, the results of our analysis are sensitive to an assumed
CO-to-H$_{\rm 2}$ conversion factor. If we assume a typical scatter of the
measurements and admit nearly twice as large a conversion factor as our
fiducial value, our analysis results in predicting the instability for the
spiral arms. More sophisticated determination of the conversion factor is
required for more accurate analysis for the (in)stability of spiral arms.
