  There are no known failures of Bounded Negativity in characteristic 0. In the
light of recent work showing the Bounded Negativity Conjecture fails in
positive characteristics for rational surfaces, we propose new characteristic
free conjectures as a replacement. We also develop bounds on numerical
characteristics of curves constraining their negativity. For example, we show
that the $H$-constant of a rational curve $C$ with at most $9$ singular points
satisfies $H(C)>-2$ regardless of the characteristic.
