Suppose I have a mining rig that draws 1,200 Watts. How many of those mining rigs can I run on a standard US home circuit?
Do I need a special outlet?
What is the risk of an electrical fire?
What other safety precautions should I take?
There isn't a single "standard" for residential circuits in the U.S. but there might be a typical circuit, such as 20 amps.
As far as plug connections, don't use light-duty extension cords; make sure they are rated for high amperage, say 20amps. Generally if you are plugging directly into an outlet then the total power drawn can be done safely up to 80% (rule of thumb) of the level the wiring is rated for.
If the circuit breaker is 20 amps and the circuit is using the proper gauge wiring for that and was wired properly, then you should be able to draw 16 amps safely from it.
16 amps * 120V = 1,920 watts.
So you could add a smaller capacity rig on the same circuit but you cannot have more than one of those 1.2 kWh rigs on what might be a "typical" residential circuit.
[Disclaimer: I'm not an electrical engineer nor an electrician but I play one on Bitcoin stackexchange.]
As a general rule of thumb, a domestic circuit has a lower or zero 'standing charge' but much higher charge per unit than a 'light industrial' one.
A friend of mine was a keen pottery enthusiast and owned a small business make and selling pots at craft fairs. He had the electrical company install a second meter and circuit (in his house) to run his electrical kiln (an oven for firing pots). This also helped when he did his accounts, as he got two separate bills, it was clear what could be offset as a business expense for tax purposes.
If you are really serious about mining maybe that would be something to consider.