We present new observational data on the small-scale structure of the Taurus molecular cloud 1 (TMC-1) in the regime of 0.02-0.04 pc and 0.04-0.6 M.. Our analysis is based on high-resolution, high-S/N, observations of an 8' x 8' area centered on the "cyanopolyyne peak" in the southeastern part of the TMC-1 ridge. The observations were made in the CCS 22 and 45 GHz transitions using NASA's Deep Space Network 70 m and 34 m telescopes at the Goldstone facility. The CCS emission in this region originates in three narrow components centered on LSR velocities of similar to 5.7, 5.9, and 6.1 km s(-1). These components each represent a separate cylindrical feature elongated along the ridge. Among the three velocity components we identified a total of 45 clumps with a typical CCS column density of similar to a few x 10(13) cm(-2), an H-2 density of similar to a few x 10(4) cm(-3), and a mass in the range of 0.04-0.6 M.. The statistical properties of these small-scale clumps are compared with those of the larger "NH3 cores" in cold clouds and "CS cores" in the hotter Orion region. The CCS clumps in TMC-1 are found to conform to Larson's scaling laws (relating observed line width to clump size) derived from the larger cores down to the small-scale regime (0.02 pc and 0.04 M.). These clumps represent a regime in which microturbulence is small, amounting to similar to 10% of the thermal pressure inside a clump. Of the 45 clumps, only five appear to be gravitationally unstable to collapse. All unbound clumps have masses less than 0.2 M., while bound clumps have masses in the range 0.15-0.6 M.. The 6.1 km s(-1) velocity feature contains all the gravitationally unstable clumps and is the most likely site for future star formation.