In the Kosher laws enumerated in Leviticus 11, The Torah mentions eight “reptiles” (Sherazim) – really “creeping things” or “vermin” – which are considered “unclean” (note that they do not appear in the parallel Kosher list in Deuteronomy 14).
The Mishnah states that capturing or wounding them on Shabbat is a culpable offense.
Since you asked, the eight are (according to the New JPS translation):
The mole, the mouse, great lizards of every variety; the gecko, the land crocodile, the lizard, the sand lizard, and the chameleon. (Lev. 11:29-30)
For other worms, snakes and insects – capturing them out of need (i.e. milking a snake for its venom) is forbidden, but to prevent biting or just to keep them from away from people is allowed.
But in any case, killing – even vermin – on Shabbat is not allowed.
Said R. Jeremiah, It is R. Eliezer. For it was taught, R. Eliezer said: He who kills vermin on the Sabbath is as though he killed a camel on the Sabbath.
a Master said: ‘The Holy One, blessed be He, sits and sustains [all creatures], from the horns of wild oxen to the eggs of vermin’
You may not care about the life of a vermin – but G-d does! On Shabbat we respect the sacredness of all life. Even pests.