Visibility of materials, body attributes, face attributes, edge attributes, and conducting nets can be toggled via the light bulb icons beneath the annotation's node in the Browser. This can be helpful for removal of visual clutter, for example to allow you to see features inside the mesh. For further discussion, and a worked-out example, see here.
For scalar fields, you can control the visibility of the surface triangles on which the fields are painted by toggling the visibility of the boundary conditions applied on the faces (the "face attributes") or edges (the "edge attributes") they represent.
For scalar and vector fields, you can also control the visibility of the surfaces or vectors by toggling the visibility of the materials from which they are constructed, or the body which they represent. A triangle/vector will be visible unless the face attribute, edge attribute, material, body attribute, or conducting net associated with it is toggled off, in which case it will be invisible. In the cases of certain fields (e.g. interception fields for 2D MICHELLE simulations), edge attribute visibility is also considered.