That's not a trivial question. The shunt resistor rating depends more upon the circuit that you are using for measuring voltage across the shunt resistor. For example if you use 0.01 ohms then full-scale voltage (at 3A) is 30mV. Does your measurement system measure accurately at 30mV? and properly amplify that up to whatever analog input range you are using?
If the answers are "yes" then at 3A the 0.01 ohm resistor will only dissipate 90mW.
If instead you do all this with a 0.1 ohm resistor then your measurement range is 300mV (easier to amplify with less noise) but now the resistor will dissipate 0.9W.
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