Whether a line sinks or not can completely depend on the type of line that you are using and the type of lure that you have. Generally without any weight attached fishing lines are quite buoyant but without anything on the end of the line you’re not going to be going much catching.
If you’re a fly fisherman you might be targeting fish that feed off the surface so in that instance you will want a lure that is going to float on the surface and a line that is not going to weigh it down. With other types of fishing you will want your lure to get to its target destination as quickly as possible so you’d have a heavier lure with a heavier line.
As we have discussed previously, the three different types of line all have different buoyancy qualities. Braided line is the most buoyant due to its light weight, followed by monofilament and finally fluorocarbon being the heaviest of the three. Despite this, clearly the lure will have a much bigger bearing on whether a line will sink or swim than the line will.