Tellurite glass microstructure fibers with a 1 microm hexagonal core were fabricated successfully by accurately controlling the temperature field in the fiber-drawing process. The diameter ratio of holey region to core (DRHC) for the fiber can be adjusted freely in the range of 1-20 by pumping a positive pressure into the holes when drawing fiber, which provides much freedom in engineering the chromatic dispersion. With the increase of DRHC from 3.5 to 20, the zero dispersion wavelengths were shifted several hundred nanometers, the cutoff wavelength due to confinement loss was increased from 1600 nm to 3800 nm, and the nonlinear coefficient gamma was increased from 3.9 to 5.7 W(-1)/m. Efficient visible emissions due to third harmonic generation were found for fibers with a DRHC of 10 and 20 under the 1557 nm pump of a femtosecond fiber laser. One octave flattened supercontinuum spectrum was generated from fibers with a DRHC of 3.5, 10 and 20 by the 1064 nm pump of a picosecond fiber laser. To the best of our knowledge, we have for the first time fabricated a hexagonal core fiber by soft glass with such a small core size, and have demonstrated a large influence of the holey region on the dispersion, nonlinear coefficient and supercontinuum generation for such fiber.