  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 secondary
structures within their bars. Some 20 +/- 3% have a separate inner bar, making
them double-barred; an identical fraction have nuclear rings, with
11^{+3}_{-2}% hosting both. The inner-bar frequency is a strong, monotonic
function 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 with
log (M_star/M_sun) > 10.5 are. The nuclear-ring frequency is a strong function
of absolute bar size: only 1^{+2}_{-1}% of bars with semi-major axes < 2 kpc
have nuclear rings, while 39^{+6}_{-5}% of larger bars do. Both inner bars and
nuclear rings are absent in very late-type (Scd--Sd) galaxies.
  Inner bar size correlates with galaxy stellar mass, but is clearly offset to
smaller sizes from the main population of bars. This makes it possible to
define "nuclear bars" in a consistent fashion, based on stellar mass. There are
eight single-barred galaxies where the bars are nuclear-bar-sized; some of
these may be systems where an outer bar failed to form, or previously
double-barred galaxies where the outer bar has dissolved. Inner bar size is
even more tightly correlated with host bar size, which is likely the primary
driver. In contrast, nuclear ring size is only weakly correlated with galaxy
mass or bar size, with more scatter in size than is true of inner bars.
