Future Terahertz communications exhibit significant potential in accommodating ultra-high-rate services. Employing extremely large-scale array antennas is a key approach to realize this potential, as they can harness substantial beamforming gains to overcome the severe path loss and leverage the electromagnetic advantages in the near field. This paper proposes novel estimation methods designed to enhance efficiency in Terahertz widely-spaced multi-subarray (WSMS) systems. Initially, we introduce three sparse channel representation methods: polar-domain representation (PD-R), multi-angular-domain representation (MAD-R), and two-dimensional polar-angular-domain representation (2D-PAD-R). Each method is meticulously developed for near-field WSMS channels, capitalizing on their sparsity characteristics. Building on this, we propose four estimation frameworks using the sparse recovery theory: polar-domain estimation (PD-E), multi-angular-domain estimation (MAD-E), two-stage polar-angular-domain estimation (TS-PAD-E), and two-dimensional polar-angular-domain estimation (2D-PAD-E). Particularly, 2D-PAD-E, integrating a 2D dictionary process, and TS-PAD-E, with its sequential approach to angle and distance estimation, stand out as particularly effective for near-field angle-distance estimation, enabling decoupled calculation of these parameters. Overall, these frameworks provide versatile and efficient solutions for WSMS channel estimation, balancing low complexity with high-performance outcomes. Additionally, they represent a fresh perspective on near-field signal processing.