Large-scale (CO)-O-18 maps show that the Taurus molecular cloud 1 (TMC-1) has numerous cores located along a ridge which extends about 12' by at least 35'. The cores traced by (CO)-O-18 are about a few arcminutes (0.1-0.2 pc) in extent, typically contain about 0.5-3 M., and are probably gravitationally bound. We present a detailed study of the small-scale fragmentary structure of one of these cores, called core D, within TMC-1 using very high spectral and spatial resolution maps of CCS and CS. The CCS lines are excellent tracers for investigating the density, temperature, and velocity structure in dense cores. The high spectral resolution, 0.008 km s(-1), data consist mainly of single-dish, Nyquist-sampled maps of CCS at 22 GHz with 45 '' spatial resolution taken with NASA's 70 m DSN antenna at Goldstone. The high spatial resolution spectral line maps were made with the Very Large Array (9 '' resolution) at 22 GHz and with the OVRO millimeter array in CCS and CS at 93 GHz and 98 GHz, respectively, with 6 '' resolution. These maps are supplemented with single-dish observations of CCS and (CCS)-S-34 spectra at 33 GHz using a NASA 34 m DSN antenna, CCS 93 GHz, (CS)-S-34 (2-1), and (CO)-O-18 (1-0) single-dish observations made with the AT&T Bell Laboratories 7 m antenna. Our high spectral and spatial CCS and CS maps show that core D is highly fragmented. The single-dish CCS observations map out several clumps which range in size from similar to 45 '' to 90 '' (0.03-0.06 pc). These clumps have very narrow intrinsic line widths, 0.11-0.25 km s(-1), slightly larger than the thermal line width for CCS at 10 K, and masses about 0.03-0.2 M.. Interferometer observations of some of these clumps show that they have considerable additional internal structure, consisting of several condensations ranging in size from similar to 10 ''-30 '' (0.007-0.021 pc), also with narrow line widths. The mass of these smallest fragments is of order 0.01 M.. These small-scale structures traced by CCS appear to be gravitationally unbound by a large factor. Most of these objects have masses that fall below those of the putative proto-brown dwarfs (less than or similar to 0.1 M.). The presence of many small gravitationally unbound clumps suggests that fragmentation mechanisms other than a purely Jeans gravitational instability may be important for the dynamics of these cold dense cores.