In this article, bifurcation buckling and snap-through stability phenomena for carbon nanotubes having a slight curvature are investigated. Carbon nanotubes are modeled as shallow arches and using nonlocal elasticity theory, the effect of small-length scale is taken into account. To obtain critical buckling loads of a sinusoidal pinned shallow arch with pinned ends resting on a Winkler foundation, the governing equations are derived and solved analytically. The possibility of bifurcation buckling and snap-through phenomena for this innovative model and the classical model is compared. It is found that the small length-scale plays a significant role in the possibility of snap-through occurrence on the arch shape carbon nanotubes. It also has important effects on their critical buckling loads.