In order to document and discuss the widespread presence in nature of the stochastic resonance phenomenon (SR), we investigate the generic double-well potential model perturbed by the a-stable Levy type noises. Despite possible infinite variance characteristics of the noise term, the SR effect still occurs and can be detected by common quantifiers used in the studies of the phenomenon. The robustness of the SR is examined by use of standard measures within a continuous and a two-state description of the system. Since the a-stable noises are characterized by a whole set of parameters, our research focuses on the analysis of the influence of noise parameters on a shape of the signal-to-noise ratio and spectral power amplification curves, revealing presence of the SR in the system at hand. Examination of the driving noise asymmetry indicates that the resonant response weakens when the symmetric noises with a decreasing value of the stability index a are applied.