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We describe the sixth worst singularity that a plane curve of degree    $$d\geqslant 5$$    d  ⩾  5      could have, using its log canonical threshold at the point of singularity. This is an extension of a result due to Cheltsov (J Geom Anal 27(3):2302–2338, 2017) wherein the five lowest values of log canonical thresholds of a plane curve of degree    $$d \geqslant 3$$    d  ⩾  3      were computed. These six small log canonical thresholds, in order, are 2 /  d ,    $$({2d-3})/{(d-1)^2}$$     (   2  d  -  3   )   /    (  d  -  1  )   2      ,    $$({2d-1})/(d^2-d)$$     (   2  d  -  1   )   /   (   d  2   -  d  )      ,    $$({2d-5})/({d^2-3d+1})$$     (   2  d  -  5   )   /   (    d  2   -  3  d  +  1   )      ,    $$({2d-3})/(d^2-2d)$$     (   2  d  -  3   )   /   (   d  2   -  2  d  )       and    $$({2d-7})/({d^2-4d+1})$$     (   2  d  -  7   )   /   (    d  2   -  4  d  +  1   )      . We give examples of curves with these values as their log canonical thresholds using illustrations.

Lowest log canonical thresholds of a reduced plane curve of degree d