Replaced or merged with Old English westen, woesten "a desert, wilderness," from the Latin word. Meanings "consumption, depletion," also "useless expenditure" are from c. 1300; sense of "refuse matter" is attested from c. 1400. Waste basket first recorded 1850.
Old English wæps, wæsp "wasp," altered (probably by influence of Latin vespa) from Proto-Germanic *wabis- (source also of Old Saxon waspa, Middle Dutch wespe, Dutch wesp, Old High German wafsa, German Wespe, Danish hveps), from PIE *wopsa-/*wospa- "wasp" (source also of Latin vespa, Lithuanian vapsa, Old Church Slavonic vosa "wasp," Old Irish foich "drone"), perhaps from root *(h)uebh- "weave" (see weave (v.)). If that is the correct derivation, the insect would be so called for the shape of its nest. Of persons with wasp-like tendencies, from c. 1500. Wasp-waist in reference to women's figures is recorded from 1870 (wasp-waisted is from 1775).
The Germanic word also existed in Old English as westan "to lay waste, ravage." Spanish gastar, Italian guastare also are from Germanic. Meaning "to lose strength or health; pine; weaken" is attested from c. 1300; the sense of "squander, spend or consume uselessly" is first recorded mid-14c.; meaning "to kill" is from 1964. Waste not, want not attested from 1778.