Nikon D5300 has a Nikon F lens mount and currently there are 257 native lenses available for this mount. Beware that Nikon D5300 doesn't have an internal focus motor so it will not auto focus with old-style screw-drive autofocus (AF) lenses. It still works well with modern Nikon AF-S, AF-I and G lenses but lenses form 1980s may not autofocus on Nikon D5300.
Nikon D5300 doesn't have a sensor based image stabilization system but 86 of these lenses already comes with optical image stabilization.
Since Nikon D5300 has a APS-C sensor, it has a focal length multiplier of 1.5× so you have to multiply lenses original focal length with this multiplier to find the Full Frame equivalent focal length of the lens when mounted on D5300.
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Nikon D5300 Lenses with Image Stabilization
Nikon D5300 Lenses with Weather Sealings