Current carrying capacity (ampacity) depends on the thickness of the leads*, which varies wildly. Cheap chinese 1/8W resistors have super thin leads (which might not even be copper), while a name-brand 2W resistor will have thick copper leads.
If you really want to know, first use a magnet to make sure they're not steel, and then use calipers to measure the diameter. You can then use any of the many online wire gauge tables (e.g. on wikipedia) to determine what gauge it is. (Or use your middle school geometry to calculate the cross section in mm
2, if you prefer metric wire sizes.) Then once you've determined this, you can refer to an
ampacity table to determine its current carrying ability.
But for 1A, you only need around 29ga wire, which is hair thin. Any name-brand component lead is going to be thick enough to carry several amps, but even the cheap Chinese ones should be thick enough for an amp or two. (The voltage is irrelevant to wire gauge, but defines how good the
insulation must be. But 30V is nothing at all. It matters once you get to mains voltages and beyond. High voltage stuff, like in the kV, is where insulation really starts to become critical.)
* and on cooling. Bare wire hanging in free air can cool better than insulated wire in a bundle of cables, for example. The better the cooling, the more current a given wire can carry. A PCB can act as a heat sink to an extent, so for example if you solder a wire to a bunch of unused perfboard pads, they'll both increase heat sinking, as well as increasing the effective thickness of the wire, reducing resistance. But for 1A that's not even needed.