Using a recently developed algorithm for generic rigidity of two-dimensional graphs, we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to $L=4096.$ We compare three different universality classes: the generic rigidity class, the connectivity class, and the generic ``braced square net''(GBSN). We analyze the spanning cluster density ${P}_{\ensuremath{\infty}},$ the backbone density ${P}_{B}$, and the density of dangling ends ${P}_{D}.$ In the generic rigidity (GR) and connectivity cases, the load-carrying component of the spanning cluster, the backbone, is fractal at ${p}_{c},$ so that the backbone density behaves as $B\ensuremath{\sim}(p\ensuremath{-}{p}_{c}{)}^{{\ensuremath{\beta}}^{\ensuremath{'}}}$ for $p>{p}_{c}.$ We estimate ${\ensuremath{\beta}}_{\mathrm{gr}}^{\ensuremath{'}}=0.25\ifmmode\pm\else\textpm\fi{}0.02$ for generic rigidity and ${\ensuremath{\beta}}_{c}^{\ensuremath{'}}=0.467\ifmmode\pm\else\textpm\fi{}0.007$ for the connectivity case. We find the correlation length exponents ${\ensuremath{\nu}}_{\mathrm{gr}}=1.16\ifmmode\pm\else\textpm\fi{}0.03$ for generic rigidity compared to the exact value for connectivity, ${\ensuremath{\nu}}_{c}=\frac{4}{3}.$ In contrast the GBSN undergoes a first-order rigidity transition, with the backbone density being extensive at ${p}_{c},$ and undergoing a jump discontinuity on reducing p across the transition. We define a model which tunes continuously between the GBSN and GR classes, and show that the GR class is typical.