May and June tend to be a quiet time of year for meteor observing. There are a variety of reasons for this : shorter nights, all night twilight in June, fairly low night-time meteor rates, plus school and university exams.
Surprises can occur however - most notably there have been several bursts of activity from the June Bootids around June 23-27 in recent years - a shower which had been thought to be defunct. Activity from the June Lyrids has also been reported on a number of occasions. Fireballs can, of course, appear at any time.
The main shower of the period is, however, the Eta Aquarids of early May. Unfortunately, this shower is difficult to observe from the UK as its radiant only rises late in the night and it is therefore only late in the night that any Eta Aquarids might be seen. At maximum in 2013, the Moon will be a waning crescent near the Aquarius-Pisces border.
Eta Aquarids are very swift meteors, often with long paths because of their low radiant, and fine persistent trains. The stream was laid down by Comet 1P/Halley, along with their autumn twin the Orionids. Halley (period around 76 years) was last at perihelion in 1986.
The most significant meteor activity during May and June actually occurs in the daytime and is thus only detectable using radio methods. The Eta Aquarids, with their late rising radiant, are in practice more of a daytime than a night time shower from the UK. Several other showers are daytime-only showers. These include the Arietids and Omicron Cetids of mid May, the Arietids and Zeta Perseids of early June and the Beta Taurids of late June.
Older meteor shower listings included an number of minor shower radiants along the ecliptic, from Virgo in early May, Scorpius in late May and Sagittarius in June. However, these are nowadays collectively referred to as the Antihelion Source, with a combined ZHR of around 3. The motion of the Antihelion Source is shown in the chart below: