We present a source catalog from the first deep hard X-ray ($E>10$ keV) survey of the Small Magellanic Cloud (SMC), the NuSTAR Legacy Survey of the SMC. We observed three fields, for a total exposure time of 1 Ms, along the bar of this nearby star-forming galaxy. Fields were chosen for their young stellar and accreting binary populations. We detected 10 sources above a 3$sigma$ significance level (4$-$25 keV) and obtained upper limits on an additional 40 sources. We reached a 3$sigma$ limiting luminosity in the 4$-$25 keV band of $sim$ $10^{35}$ erg s$^{-1}$, allowing us to probe fainter X-ray binary (XRB) populations than has been possible with other extragalactic NuSTAR surveys. We used hard X-ray colors and luminosities to constrain the compact-object type, exploiting the spectral differences between accreting black holes and neutron stars at $E>10$ keV. Several of our sources demonstrate variability consistent with previously observed behavior. We confirmed pulsations for seven pulsars in our 3$sigma$ sample. We present the first detection of pulsations from a Be-XRB, SXP305 (CXO J005215.4$-$73191), with an X-ray pulse period of $305.69pm0.16$ seconds and a likely orbital period of $sim$1160-1180 days. Bright sources ($gtrsim 5times 10^{36}$ erg s$^{-1}$) in our sample have compact-object classifications consistent with their previously reported types in the literature. Lower luminosity sources ($lesssim 5times 10^{36}$ erg s$^{-1}$) have X-ray colors and luminosities consistent with multiple classifications. We raise questions about possible spectral differences at low luminosity between SMC pulsars and the Galactic pulsars used to create the diagnostic diagrams.
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