Both explosive and incendiary bullets were developed in WW1 for attacking observation balloons, airships and aircraft. This was strictly against the St Petersburg Declaration of 1868 which banned the use of explosive bullets (in fact, all explosive projectiles of less than 400 grams weight), but after the war was over it was generally agreed that since these bullets were not intended for use against people, it was OK to use them in aircraft and AA guns.

Not much use was made of explosive rifle-calibre bullets in WW2 because the quantity of HE they could contain was too small to be worthwhile. The Japanese perhaps made the most use, getting in more HE by omitting any kind of fuze and just putting highly sensitive PETN behind a thin nose cap (real fun to handle...). Incendiary bullets, most of which burst into flame on impact, were common though.

Nobody made any regular use of HE/incendiary bullets for ground use. That doesn't mean to say that someone couldn't have got hold of some unofficially...

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Tony Williams: Military gun and ammunition website