Geodetic/astrometric very long baseline interferometry (VLBI) has been routinely observing using various global networks for 40 yr, and it has produced more than 10 million baseline group delay, phase, and amplitude observables. These group delay observables are analyzed worldwide for geodetic and astrometric applications, for instance, to create the International Celestial Reference Frame (ICRF). The phase and amplitude observables are used in this paper, by means of closure analysis, to study intrinsic source structures and their evolution over time. The closure amplitude rms, CARMS, indicating how far away a source is from being compact in terms of morphology, is calculated for each individual source. The overall structure-effect magnitudes for 3417 ICRF radio sources are quantified. CARMS values larger than 0.3 suggest significant source structures and those larger than 0.4 indicate very extended source structures. The 30 most frequently observed sources, which constitute 40% of current geodetic VLBI observables, are studied in detail. The quality of ICRF sources for astrometry is evaluated by examining the CARMS values. It is confirmed that sources with CARMS values larger than 0.30 can contribute residual errors of about 15 ps to geodetic VLBI data analysis and those with the CARMS values larger than 0.4 generally can contribute more than 20 ps. We recommend CARMS values as an indicator of the astrometric quality for the ICRF sources and the continuous monitoring of the ICRF sources to update CARMS values with new VLBI observations as they become available.