From my understanding, standards such as WMS, TMS etc are mainly used to fetch raster layers from a map server.
What are the popular options for serving vector layers? Can it be tiled or not tiled as well?
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WFS. Web Feature Service would be the vector equivalent of WMS.
For tiles vectors there is vector tiles, which can be served through TMS. For example with GeoServer: https://docs.geoserver.org/latest/en/user/extensions/vectortiles/tutorial.html