An automatic classification method to efficiently discriminate the types of discontinuous breath sounds (DBSs), i.e., fine crackles (FCs), coarse crackles (CC), and squawks (SQ), is presented in this paper. Using the lacunarity of the acquired DBS, the proposed classification method, namely LAC, introduces a texture-based approach that captures the differences in the distribution of FC, CC, and SQ across the breathing cycle, which may lead to more accurate characterization of the pulmonary acoustical changes due to the related pathology. Prior to the lacunarity analysis, wavelet-based denoising of DBS is employed to eliminate effects of the vesicular sound (background noise) to DBS oscillatory pattern. LAC analysis builds its classification power both upon the use of lacunarity at an optimum scale and the approximation of its trajectory across an optimum range of scales using a three-parameter hyperbola model. LAC is applied to 363 DBS corresponding to 25 cases included in four lung sound databases. Results show that LAC efficiently classifies the three DBS categories in the comparison groups of FC-CC, FC-SQ (both with mean accuracy of 100%), CC-SQ (mean accuracy of 99.62%-100%), and FC-CC-SQ (mean accuracy of 99.75%-100%). When compared to other classification tools, LAC seems quite attractive, since, without employing high computational complexity, it results in high classification accuracy. Moreover, LAC introduces a "texture" concept in the analysis of breath sounds, something that strongly relates to the perception of the bioacoustic signals by the physician. Due to its simplicity, LAC could be implemented in a real-time context and be used in clinical medicine as a module of an integrated intelligent patient evaluation system.