Can I take the cover off my fusebox to attach an energy monitor?
I would find some way to attach the measuring clamp around the live feed near the meter.

My house, as many do, has an external wall-box flush-mounted into an external brick wall. It used to look like this




Mine is a bit unusual in that the supply is three-phase rather than the more usual single-phase but the general layout is the same. 


At bottom left is the incoming unmetered supply terminated at 100A (230V) supply fuses. You are not permitted to touch these.
Neutral and Live (usually one but three in my case) wires feed the meter.
You can see the after-meter live wire (marked with red tape) disappears off into the wall at top left.


There should always be enough room to attach a current sensing clamp over that wire. The wires are very stiff but you should be able to move them enough to attach a typical clamp around the live wire safely.

Inside my house, on the opposite side of the same wall, is a separate wall-mounted "consumer unit" inside a cupboard. These are often called a fuse-box although these distribution panels now contain miniature circuit breakers (MCBs) rather than fuses.



There is no way to attach a current sensing clamp here and I would advise against opening up the panel to find the incoming live feed. You would need to isolate the supply into the building and there is no facility for householders to do this - I believe you are expected to contact the electricity supply company if you need to isolate anything before the main switch on the consumer unit.

The current-sense clamps supplied with energy monitors are intended to be used in the exterior meter box.