Maximum rating of resistor
You read the data sheet. 

The maximum steady-state current depends on the resistor rating and (at least above some minimum) the ambient temperature. 

Surge ratings should be in the data sheet.
Fusible resistors are designed to fail “open” in a fairly predictable manner, without undue flame, smoke and other drama. 

I would think that a fusible wire wound resistor would be more appropriate than a film type for an application with relatively long duration surges.