According to the sunnah of Nabi SAW, kohl/surma is applied before going to sleep. Therefore, applying kohl/surma in public is not according to Sunnah.
It comes in a hadith that 'A woman herself is 'aurah. When she goes out, Syaithaan beautifies her (uses her to become a means of fitnah)'.
A woman leaving her house perfumed and adorned even with her husband's permission is regarded as a major sin. ('Umdatus Saalik pg: 983)
Thus, it does not befit a woman to apply any form of make-up in public as it is not permissible and sinful as a woman should always observe hijab when in public.
