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Using a volume- and mass-limited (D < 30 Mpc, log (M_star/M_sun) $\geq 9.75$)sample of 155 barred S0-Sd galaxies, I determine the fraction with secondarystructures within their bars. Some 20 +/- 3% have a separate inner bar, makingthem double-barred; an identical fraction have nuclear rings, with11^{+3}_{-2}% hosting both. The inner-bar frequency is a strong, monotonicfunction of stellar mass: only 4^{+3}_{-2}% of barred galaxies with log(M_star/M_sun) = 9.75-10.25 are double-barred, while 47 +/- 8% of those withlog (M_star/M_sun) > 10.5 are. The nuclear-ring frequency is a strong functionof absolute bar size: only 1^{+2}_{-1}% of bars with semi-major axes < 2 kpchave nuclear rings, while 39^{+6}_{-5}% of larger bars do. Both inner bars andnuclear rings are absent in very late-type (Scd--Sd) galaxies.  Inner bar size correlates with galaxy stellar mass, but is clearly offset tosmaller sizes from the main population of bars. This makes it possible todefine "nuclear bars" in a consistent fashion, based on stellar mass. There areeight single-barred galaxies where the bars are nuclear-bar-sized; some ofthese may be systems where an outer bar failed to form, or previouslydouble-barred galaxies where the outer bar has dissolved. Inner bar size iseven more tightly correlated with host bar size, which is likely the primarydriver. In contrast, nuclear ring size is only weakly correlated with galaxymass or bar size, with more scatter in size than is true of inner bars.

The Frequency and Sizes of Inner Bars and Nuclear Rings in Barred Galaxies and Their Dependence on Galaxy Properties