In this paper we introduce one type of generalized dispersive Camassa-Holm model and make its singularity analysis. We prove that the model is Painlevé integrable by an alternative WTC-Kruskal test and obtain the Painlevé-Bcklund systems and the Bcklund transformation. Many new types of regular soliton, singular soliton, kink soliton, compacton and anti-compacton are explored. Particularly, we have found singular structures of periodic cuspon waves in kink solitons, which occur in their central regions. Based on the regular solitonic system, we do Bcklund transformation and obtain three sorts of singular solitons, namely the periodic blow-up wave with hump structure, kink soliton for the blow-up wave structure and the compacton.