Dirt or blow by in filter?

What engine?
Piston ported 2 strokes naturally have a certain amount of blow back through the carburetor. Oil residue in the air filter element is normal. The oil then collects road dust and dirt. One reason to maintain the air filter regularly. Oil on the outside of the carburetor usually means you have a leak around the float bowl gasket or the float level is too low and you're flooding the carb. Crud in the float needle valve will also cause a high fuel level which can drain out of the tickler button and the inlet of the carb. A cracked or damaged float will also cause the same problem.

If you have a 4 stroke engine you shouldn't be seeing this problem. It could be a early sign of blow by in the piston rings and leaking valves.

Tom

Not carbon from exhaust but certainly signs of leaks. As I said above, the dirt is road dust sticking to oil residue. If you're seeing that around the spark plug, your plug might be too loose. Some kit instructions will tell you to "finger tighten the plug". That's just nonsense. It needs to be a lot tighter than that, about 80 to 100 inch pounds will be good.

As for the carburetor, I explained that above. You might want to check where the fuel line attaches to the carb. Your fuel line might be leaking slightly.

Tom