If you'd like to see bits of debris shed by Halley's Comet, then head out before dawn during the next few days.
Only one major meteor stream, the one shed by Comet 1P/Halley, intersects Earth's orbit in two places such that both resulting showers are visible at night. In October we call them the Orionids, and in May they're the Eta Aquariids.
Both showers run for several days and have a reputation for meteors that are swift and relatively faint. But in May we apparently pass nearer to the center of the stream; the Eta Aquariid shower is roughly three times as strong as the Orionid shower, and it lasts longer too.
This year the Eta Aquariid shower is set to peak on the mornings of May 5th and 6th, when its meteors will be untroubled by moonlight.
The Eta Aquariid meteor shower is usually the year's best for those living in the Southern Hemisphere, with perhaps 60 "shooting stars" visible per hour before dawn to an observer with ideal conditions. Meteor watchers in northern latitudes see fewer, and those north of 40° or 45° see hardly any, because the shower's radiant (at the Water Jug or Jar, the head of Aquarius) is still low in the east-southeast as dawn brightens.
On the other hand, when a shower's radiant is low, the few meteors that do show up tend to be long, dramatic Earthgrazers passing far across the sky. That seems somehow fitting for dusty bits cast into space by none other than Halley's Comet.