Textures on procedurally generated mesh have lines in between

I just started on procedurally generated meshes and was going along great until I noticed this nagging problem that I really hope someone can shed some light on. I have a giant quad mesh made up of many little quads, and when I UV map some textures to these quads from a big tilesheet texture lines appear in between each of the little quads. The amount and intensity of the lines change when I change the Wrap Mode and Filter Mode of the tilesheet, though none of the options gets rid of them all, which is what I really want.

The tilesheet is here (only using the bottom left tile) and the UV mapping code is below:

 for(i = 0;i < p.quadCount;i++) {
     uv[0 + (i*4)] = new Vector2(0, 0);
     uv[1 + (i*4)] = new Vector2(1.0/p.tileSheetX, 0);
     uv[2 + (i*4)] = new Vector2(1.0/p.tileSheetX, 1.0/p.tileSheetY);
     uv[3 + (i*4)] = new Vector2(0, 1.0/p.tileSheetY);
 }

This for loop loops through every quad on the mesh and UV maps a section of the tilesheet to the four corners of every quad, with p.tileSheetX and p.tileSheetY being how many tiles are on the tilesheet (in this case p.tileSheetX = 8 and p.tileSheetY = 7). If I change this to the below:

 for(i = 0;i < p.quadCount;i++) {
     uv[0 + (i*4)] = new Vector2(0.01, 0.01);
     uv[1 + (i*4)] = new Vector2(1.0/p.tileSheetX - 0.01, 0.01);
     uv[2 + (i*4)] = new Vector2(1.0/p.tileSheetX - 0.01, 1.0/p.tileSheetY - 0.01);
     uv[3 + (i*4)] = new Vector2(0.01, 1.0/p.tileSheetY - 0.01);
 }

then there are no more lines unless you zoom very far out from the mesh, though if you go to 0.001 instead of 0.01 the lines reappear. This seems messy and doesn't appear very scalable for large tilesheets so I'd like a better fix.

Wrap - Clamp, Filter - Point

Wrap - Repeat, Filter - Point

Wrap - Clamp, Filter - Bilinear

Wrap - Repeat, Filter - Bilinear