The best site for up-to-date comet viewing information is Seichi Yosida’s comet site here: www.aerith.net/comet/weekly/current.html.
The screenshot below is the current posting from that site (used with permission) and shows comets above magnitude 12 or so that are presently in the sky. More details (including finder charts) about one or more of the comets (some possibly within binocular range) can be found at the above link and at other sites like Sky&Telescope.
The diagram from Starry Night below shows the path each of the above comets takes, and the dates are marked on the tracks. For example, on May 22, ATLAS Y4 is just at the horizon, ATLAS Y1 is in the bowl of the Big Dipper and PanSTARRS is about where M81 and M82 are located.
The Moon-free periods in the next three months are March 20 to 30, April 15 to 25, and May 12 to 26 or so. Comet Y4 ATLAS is due to be brightest in the latter interval but it also is dropping towards the horizon at this time.
Update (April 4/2020): Atlas Y4 is up to magnitude 7 or so and brightening. When the moon becomes last quarter and then new, and out of the sky for a while, it will be a good time to go looking. Approximate dates are April 10-24. John H. took an image a week or so ago and it was barely detectable, but it is supposed to be brightening to naked eye visibility later in April/early May.
C/2017 T2 (PanSTARRS)
The PanSTARRS search telescope is constantly turning up new “fuzzies” and this comet has now been seen by one of our BAS members (at least once). Note that it starts out near Cassiopeia in early March and proceeds to climb in the sky towards Ursa Major so it is a circumpolar comet visible all night. It is not expected to brighten more than to about magnitude 8 but it will be in the sky at that brightness for several months.
C/2019 Y1 (ATLAS)
This comet is also circumpolar and visible all night long after the start of April or so. It travels about twice the sky that T2 PanSTARRS does in the same time interval. It is not inherently bright and is not expected to get much brighter than magnitude 10 or maybe 9.5 by April. It also will be visible into May.
A brand new comet has been spotted by the Hubble Space Telescope, and remarkably, even though it is still far enough away from the Sun that its water ice would be as hard as rock, it is already showing signs of activity. That new comet -- called C/2017 K2 (PANSTARRS) or K2 for short -- is the farthest active comet astronomers have ever seen. It is at a distance of 2.4 billion kilometres from the Sun which is beyond the orbit of Saturn. Discovered in May 2017 using Hawaii's Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), astronomers used the Hubble Space Telescope to get a better look at K2. Hubble revealed that the comet's coma -- the gassy, dusty "atmosphere" that surrounds the comet's solid nucleus -- was already around 115,000 km across. That's roughly the diameter of the planet Saturn! The closest approach to the Sun is predicted for 2022.
Read more about it here.
Tim Reyes writes for EarthSky.org. One of his recent excellent articles is When is Our Next Great Comet? An interesting read! [Maybe it will be Comet C/2017 K2 PanSTARRS , see above].