1) Classic Knotting problem: Classify embeddings of circle into 3D Euclidean space up to isotopy. http://en.wikipedia.org/wiki/Knot_theory

2) General topological knotting problem: Classify embeddings of one topological space into another up to isotopy. http://www.map.him.uni-bonn.de/index.php/High_codimension_embeddings:_classification

3) General knotting problem: Classify embeddings of some kind of one structure into another up to whatever equivalence relation is considered interesting.

Question: What examples have been studied of classifying embeddings of the third kind ?

For example has there been much work done on classifying embeddings of one metric space into another up to isometry ?

Edit: To try to make this more specific and more like a knotting problem - it shouldn't be any old equivalence relation but one that arises from morphisms of the larger space to itself and finding that in doing so, some embeddings can be transformed into each other and others cannot.

The more like a knotting problem a situation is the better, for example if there are an infinite number of equivalence classes, some measure of complexity with the simplest class being termed the unknot.