The bounded negativity conjecture predicts that if \(X\) is a smooth complex projective surface, then there is a bound \(B_X\) such that every reduced curve \(C \subset X\) satisfies \(C^2 \geq B_X\) and one may take \(C\) irreducible by work of T. Bauer et al. [Duke Math. J. 162, No. 10, 1877–1894 (2013; Zbl 1272.14009)]. There are counterexamples in every positive characteristic, even for rational surfaces [R. Cheng and R. van Dobben de Bruyn, J. Reine Angew. Math. 783, 217–226 (2022; Zbl 1483.14019)], which implies that there are plane curves with singular points of high multiplicity compared to their degrees. For a reduced plane curve \(C \subset \mathbb P^2\) having \(r > 0\) singularities \(p_1, \dots, p_r\) (possibly infinitely near) with respective multiplicities \(m_i\), let \(\displaystyle H(C) = \frac{d^2 - \sum m_i^2}{r}\). Then a lower bound on \(H(C)\) over all reduced curves \(C\) implies the negativity conjecture. For such a curve \(C\), the authors define \((d,m_1, \dots, m_r)\) to be a multiplicity sequence. The authors pose several conjectures and questions about multiplicity sequences and possibly bounds on \(H(C)\), proving various interconnections between them. The paper has many examples and remarks and can be used as a survey paper on this approach to the negativity conjecture.