The distribution of star formation in galactic bars as seen with H$alpha$ and stacked GALEX UV imaging

We investigate the spatial distribution of star formation (SF) within bars of nearby disk galaxies (inclination $< 65^{circ}$) from the S$^4$G survey. We use archival GALEX far- and near-UV imaging for 772 barred galaxies. We also assemble a compilation of continuum-subtracted H$alpha$ images for 433 barred galaxies, of which 70 are produced by ourselves from ancillary photometry and MUSE/CALIFA IFU data cubes. We employ two complementary approaches: i) the analysis of bar/disk stacks built from co-added UV images of hundreds of galaxies; and ii) the classification of the morphology of ionised regions in galaxies into three main SF classes: A) only circumnuclear SF, B) SF at the bar ends, but not along the bar, and C) SF along the bar. Lenticular galaxies typically belong to SF class A: this is probably related to bar-induced SF quenching. The distribution of SF class B peaks for early- and intermediate-type spirals: this most likely results from the interplay of gas flow, shocks, and enhanced shear in centrally concentrated galaxies with large bar amplitudes. Late-type galaxies are mainly assigned to SF class C: we argue that this is a consequence of low shear. In bar stacks of spirals, the UV emission traces the stellar bars and dominates on their leading side, as witnessed in simulations. For early-types, the central UV emission is $sim$0.5 mag brighter in strongly barred galaxies, relative to their weakly barred counterparts: this is related to the efficiency of strong bars sweeping the disk gas and triggering central starbursts. We also show that the distributions of SF in inner ringed galaxies are broadly the same in barred and non-barred galaxies, including a UV/H$alpha$ deficit in the middle part of the bar: this hints at the effect of resonance rings trapping gas. Distinct distributions of SF within bars are reported in galaxies of different morphological types (Abridged).