Are static variables in Objective-C methods shared across instances?

It's the same as a static variable in C; the instances will share the variable. If you want each instance to have its own copy, you want an instance variable (declared in the @interface block).

Static locals are shared between method calls AND instances. You can think of them as globals which are visible only inside their methods:

- (void) showVars {
    int i = 0;
    static int j = 0;
    i++; j++;
    NSLog(@"i = %i ; j = %i", i, j);
}

[...]

[obj1 showVars];
[obj2 showVars];
[obj1 showVars];
[obj2 showVars];

Above calls on 2 different instances will output:

i = 1 ; j = 1
i = 1 ; j = 2
i = 1 ; j = 3
i = 1 ; j = 4