We report on a study of the alpha-decay fine structure and the associated E alpha-E gamma correlations in the decays of 171,172Os and 171,172,174Ir. In total, 13 new alpha-decay energy lines have been resolved, and three new gamma-ray transitions have been observed following the new decay branches to 168Re and 167W. The weak alpha-decay branch from the bandhead of the nu i13/2 band in 171Os observed in this work highlights an unusual competition between alpha, beta, and electromagnetic decays from this isomeric state. The nucleus 171Os is therefore one of few nuclei observed to exhibit three different decay modes from the same excited state. The nuclei of interest were produced in 92Mo(83Kr, xpyn) fusion-evaporation reactions at the Accelerator Laboratory of the University of Jyvaskyla, Finland. The fusion products were selected using the gas-filled ion separator RITU and their decays were characterized using an array of detectors for charged particles and electromagnetic radiation known as GREAT. Prompt gamma-ray transitions were detected and correlated with the decays using the JUROGAM II germanium detector array surrounding the target position. Results obtained from total Routhian surface (TRS) calculations suggest that alpha-decay fine structure and the associated hindrance factors may be a sensitive probe of even relatively small shape changes between the final states in the daughter nucleus.