Start with a sharp face
Choose a frame where the eyes and face are in focus. A slightly soft tail is often acceptable; a blurred face rarely is. Natural side light can reveal individual fur strands, while dim indoor light and motion blur blend fur into the background.
Contrast matters around the outside, not just in the center. A black dog on a dark sofa or a pale cat against a bright window gives the mask little separation. Move your viewpoint or choose another frame before attempting a detailed correction.
Review the whole silhouette first
Before zooming in, check that the automatic result includes the full ears, chest, paws, and tail. Restore any large missing area with a larger brush. Remove obvious floor, blanket, or furniture regions with Erase. This gets the shape right before you spend time on individual edges.
Refine fur at a useful scale
Pinch to zoom and reduce the brush size. Work in short strokes parallel to the edge. Restore into fur that should remain; erase pockets of background between longer strands. Frequent small corrections are easier to undo than a long stroke around the whole animal.
Whiskers are a special case
Fine translucent whiskers can fall below the detail available in the segmentation mask. Keeping the face and muzzle clean is usually more important than preserving every whisker. A solid white or dark background can also hide small edge problems that stand out on transparency.
Pick the background after refinement
Test the result over white, black, and a contrasting color. Each reveals different halos. If the final use needs transparency, fix the most obvious fringe while previewing against more than one color, then return to Clear and export PNG.