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I'm making a dagger in a half moon shape. I got this far (see picture below), but I really want to make an edge loop on the sides (not the sharp edge of the blade, but the side), but I only get vertices suggestions on the edges and not for the entire length.

I used a cylinder to make this and used the boolean modifier with another cylinder.

enter image description here

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    $\begingroup$ Your title says "no ngons" but from your picture it looks a little like there are ngon faces at the tips of the half moon? If you are trying to cut an edge loop using Ctrl-R those (apparent) ngons may cause an issue. Perhaps posting your blend file would help? $\endgroup$ – risingfall Jan 30 '18 at 18:57
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Basically like this but you can shorten it up by moving the vertices on the bottom or top.If I may add, it might be due to the edges on the sides converging into a point on the left and right vertices at the end of the crescent. IF I am not mistaken the edge loops needs to be in between two vertices. Try extruding the vertices on both sides and using a loop cut. That should work.

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In such cases use Ctrl+RMB 🖱 to select the shortest path. Make Edge Loop works best when the edge is not segmented/subdivided.

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The reason your edge loops aren't continuing as you would like is because you've created n-poles towards the end of your crescent. You've done this in order to maintain all-quad topology, but all-quad topology is only useful as long as it lets you continue edge loops as you want to. The corner where the three highlighted lines intersect, as far as the topology is concerned, is exactly identical to the corner of a cube. Blender doesn't know which edge to continue the edge loop down, so it stops there.

enter image description here

Option 1: Take the Triangle

In this case, you can just leave triangles there. They're not really going to screw up a subdivision surface modifier, and for the most part they will let you continue to add edge loops along the geometry as needed.

enter image description here

Option 2: Better Quad Geometry

If you really want to keep to quads though, you simply need to add some more geometry which puts your quads in the right places.

This arrangement preserves quads, but doesn't put those inconvenient poles along the edge loops you want to select:

enter image description here

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