A cell already has its number of lines (rule out the rest), or needs every undecided edge to reach it (draw them all).
Dot rulesT2
The loop passes through each dot 0 or 2 times. A dot with two lines rules out its other edges; a dot with one line and one open edge must continue; a lone open edge at an empty dot is ruled out.
Corner reasoningT3
Short case analysis at a cell's corner. A 3 whose corner is sealed from outside must use both corner edges; a 1 must use neither. A line entering a corner of a 1 exhausts it; entering a 3 forces the two far edges; entering a 2 must exit through the opposite corner, and a sealed corner of a 2 pushes the loop out through its neighbors.
3-3 patternsT4
Two 3s side by side force the shared and outer parallel edges (and stop the shared edge from extending); two 3s on a diagonal force both outer corners.
No early loopT5
An edge that would join the two ends of an existing path may not be drawn if closing the loop there would leave the rest of the puzzle unfinishable.
One-edge trialT6
Suppose a single edge is a line (or a ×) and follow the counting rules; if that quickly contradicts, the edge must be the opposite. The workhorse of hard slitherlinks.