We report the quantitative characterization by synchrotron soft X-ray spectroscopy of 31 potential impact features in the aerogel capture medium of the Stardust Interstellar Dust Collector. Samples were analyzed in aerogel by acquiring high spatial resolution maps and high energy-resolution spectra of major rock-forming elements Mg, Al, Si, Fe, and others. We developed diagnostic screening tests to reject spacecraft secondary ejecta and terrestrial contaminants from further consideration as interstellar dust candidates. The results support an extraterrestrial origin for three interstellar candidates: I1043,1,30 (Orion) is a 3 pg particle with Mg-spinel, forsterite, and an iron-bearing phase. I1047,1,34 (Hylabrook) is a 4 pg particle comprising an olivine core surrounded by low-density, amorphous Mg-silicate and amorphous Fe, Cr, and Mn phases. I1003,1,40 (Sorok) has the track morphology of a high-speed impact, but contains no detectable residue that is convincingly distinguishable from the background aerogel. Twenty-two samples with an anthropogenic origin were rejected, including four secondary ejecta from impacts on the Stardust spacecraft aft solar panels, nine ejecta from secondary impacts on the Stardust Sample Return Capsule, and nine contaminants lacking evidence of an impact. Other samples in the collection included I1029,1,6, which contained surviving solar system impactor material. Four samples remained ambiguous: I1006,2,18, I1044,2,32, and I1092,2,38 were too dense for analysis, and we did not detect an intact projectile in I1044,3,33. We detected no radiation effects from the synchrotron soft X-ray analyses; however, we recorded the effects of synchrotron hard X-ray radiation on I1043,1,30 and I1047,1,34