Beam-target double-spin asymmetries and target single-spin asymmetries in exclusive ${\ensuremath{\pi}}^{+}$ and quasiexclusive ${\ensuremath{\pi}}^{\ensuremath{-}}$ electroproduction were obtained from scattering of 1.6- to 5.7-GeV longitudinally polarized electrons from longitudinally polarized protons (for ${\ensuremath{\pi}}^{+}$) and deuterons (for ${\ensuremath{\pi}}^{\ensuremath{-}}$) using the CEBAF Large Acceptance Spectrometer (CLAS) at Jefferson Lab. The kinematic range covered is $1.1<W<2.6$ GeV and $0.05<{Q}^{2}<5\phantom{\rule{0.28em}{0ex}}{\mathrm{GeV}}^{2}$, with good angular coverage in the forward hemisphere. The asymmetry results were divided into approximately 40 000 kinematic bins for ${\ensuremath{\pi}}^{+}$ from free protons and 15 000 bins for ${\ensuremath{\pi}}^{\ensuremath{-}}$ production from bound nucleons in the deuteron. The present results are found to be in reasonable agreement with fits to previous world data for $W<1.7$ GeV and ${Q}^{2}<0.5\phantom{\rule{0.28em}{0ex}}{\mathrm{GeV}}^{2}$, with discrepancies increasing at higher values of ${Q}^{2}$, especially for $W>1.5$ GeV. Very large target-spin asymmetries are observed for $W>1.6$ GeV. When combined with cross-section measurements, the present results can provide powerful constraints on nucleon resonance amplitudes at moderate and large values of ${Q}^{2}$, for resonances with masses as high as 2.3 GeV.