We present a sample of 1483 sources that display spectral peaks between 72 MHz and 1.4 GHz, selected from the GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) survey. The GLEAM survey is the widest fractional bandwidth all-sky survey to date, ideal for identifying peaked-spectrum sources at low radio frequencies. Our peaked-spectrum sources are the low-frequency analogs of gigahertz-peaked spectrum (GPS) and compact-steep spectrum (CSS) sources, which have been hypothesized to be the precursors to massive radio galaxies. Our sample more than doubles the number of known peaked-spectrum candidates, and 95% of our sample have a newly characterized spectral peak. We highlight that some GPS sources peaking above 5 GHz have had multiple epochs of nuclear activity, and we demonstrate the possibility of identifying high-redshift (z > 2) galaxies via steep optically thin spectral indices and low observed peak frequencies. The distribution of the optically thick spectral indices of our sample is consistent with past GPS/CSS samples but with a large dispersion, suggesting that the spectral peak is a product of an inhomogeneous environment that is individualistic. We find no dependence of observed peak frequency with redshift, consistent with the peaked-spectrum sample comprising both local CSS sources and high-redshift GPS sources. The 5 GHz luminosity distribution lacks the brightest GPS and CSS sources of previous samples, implying that a convolution of source evolution and redshift influences the type of peaked-spectrum sources identified below 1 GHz. Finally, we discuss sources with optically thick spectral indices that exceed the synchrotron self-absorption limit.