Nucleic acid nanoparticles (NANPs) are an emerging class of programmable structures with tunable shape and function. Their promise as tools for fundamental biophysics studies, molecular sensing, and therapeutic applications necessitates methods for their detection and characterization at the single-particle level. In this work, we study electrophoretic transport of individual ring-shaped and cube-shaped NANPs through solid-state nanopores. In the optimal nanopore size range, the particles must deform to pass through, which considerably increases their residence time within the pore. Such anomalously long residence times permit detection of picomolar amounts of NANPs when nanopore measurements are carried out at a high transmembrane bias. In the case of a NANP mixture, the type of individual particle passing through nanopores can be efficiently determined from analysis of a single electrical pulse. Molecular dynamics simulations provide insight into the mechanical barrier to transport of the NANPs and corroborate the difference in the signal amplitudes observed for the two types of particles. Our study serves as a basis for label-free analysis of soft programmable-shape nanoparticles.