  We study slope stability of smooth surfaces and its connection with
exceptional divisors. We show that a surface containing an exceptional divisor
with arithmetic genus at least two is slope unstable for some polarisation. In
the converse direction we show that slope stability of surfaces can be tested
with divisors, and prove that for surfaces with non-negative Kodaira dimension
any destabilising divisor must have negative self-intersection and arithmetic
genus at least two. We also prove that a destabilising divisor can never be
nef, and as an application give an example of a surface that is slope stable
but not K-stable.
