VSAM (Virtual Storage Access Method) supports four main file organization types: KSDS (Key Sequenced Data Set) for indexed access, ESDS (Entry Sequenced Data Set) for sequential-only access, RRDS (Relative Record Data Set) for direct access via relative record number, and LDS (Linear Data Set). COBOL can work with KSDS, ESDS, and RRDS, which are the primary VSAM file types used in mainframe applications.