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As the earliest meshless method, Smoothed Particles Hydrodynamics (SPH) has been applied in solid dynamics because of its great potentials in simulating extremely large deformations. However, the numerical instability of SPH is still a severe problem, especially tensile instability and pairing instability can lead to some unphysical cracks when the solid material is stretched or compressed. At present, no method exists to completely avoid these instabilities, although a few corrections like Artificial Viscosity, Artificial Stress, CSPH or the Godunovtype SPH have been proposed. The base of SPH formulation uses a kernel function for numerical approximations. Some literatures demonstrate mathematically that the types of the kernel function directly influences the stability of SPH method. In this paper, we study the stability of SPH with the different kinds of kernel functions including traditional types and several new ones proposed recently. Combined with some corrected techniques, the suitability of these kernels in SPH method is discussed in solid dynamic problems like bending deformation in elastic beam and elastic-plastic deformation in impact problems.