Local-Group galaxies provide access to samples of X-ray source populations of whole galaxies. The XMM-Newton survey of the Small Magellanic Cloud (SMC) completely covers the bar and eastern wing with a 5.6 deg^2 area in the (0.2-12.0) keV band. To characterise the X-ray sources in the SMC field, we created a catalogue of point sources and sources with moderate extent. Sources with high extent (>40) have been presented in a companion paper. We searched for point sources in the EPIC images using sliding-box and maximum-likelihood techniques and classified the sources using hardness ratios, X-ray variability, and their multi-wavelength properties. The catalogue comprises 3053 unique X-ray sources with a median position uncertainty of 1.3 down to a flux limit for point sources of ~10^-14 erg cm^-2 s^-1 in the (0.2-4.5) keV band, corresponding to 5x10^33 erg s^-1 for sources in the SMC. We discuss statistical properties, like the spatial distribution, X-ray colour diagrams, luminosity functions, and time variability. We identified 49 SMC high-mass X-ray binaries (HMXB), four super-soft X-ray sources (SSS), 34 foreground stars, and 72 active galactic nuclei (AGN) behind the SMC. In addition, we found candidates for SMC HMXBs (45) and faint SSSs (8) as well as AGN (2092) and galaxy clusters (13). We present the most up-to-date catalogue of the X-ray source population in the SMC field. In particular, the known population of X-ray binaries is greatly increased. We find that the bright-end slope of the luminosity function of Be/X-ray binaries significantly deviates from the expected universal high-mass X-ray binary luminosity function.
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