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High-resolution spectroscopy has been performed on the ground-state hyperfine transitions in trapped 201Hg+ ions as part of a program to investigate the viability of 201Hg+ for clock applications. Part of the spectroscopy work was directed at magnetic-field-sensitive hyperfine lines with ??mF =0, which allow accurate Doppler-free measurement of the magnetic field experienced by the trapped ions. Although it is possible to measure Doppler-free magnetic-field-sensitive transitions in the commonly used clock isotope, 199Hg+, it is more difficult. In this paper, we discuss how this 201Hg+ feature may be exploited to produce a more stable clock or one requiring less magnetic shielding in environments with magnetic field fluctuations far in excess of what is normally found in the laboratory. We have also determined that in discharge-lamp-based trapped mercury ion clocks, the optical pumping time for 201Hg+ is about 3 times shorter than that of 199Hg+ This can be used to reduce dead time in the interrogation cycle for these types of clocks, thereby reducing the impact of local oscillator noise aliasing effects.