I always had Brum down as one of the snowier locations in the southern half of the country. Maybe my mind is playing tricks but I seem to recall countless times when they've had snow and Liverpool has had nothing. The inland location and relatively high altitude (for a city) certainly work in Birmingham's favour when it comes to marginal situations, which annoyingly tend to lead to my location just missing out: given that snow isn't all that common for most of the country, it's these marginal setups which are most common for snowfall. Very cold synoptics destroy marginality and remove Birmingham's advantage for snowfall, but of course this scenario isn't as common.
Just consulted "The Weather Of Britain" by Robin Stirling, which states that Birmingham had 75 days of snow-cover in the winter of 1962/3 compared with 45 at Kew and 36 at Manchester - it's also higher than Buxton managed (74 days). Since 1917/8. Birmingham has had two snowless winters (1924/5 and 1933/4) - the book was published in 1997, but I don't think there has been a completely snow-free winter there since.
I'm not sure if Birmingham has a particulary impressive record regarding snow depths, but it would seem that it is a pretty good place for frosts and lying snow days. You can compare it on a larger scale with Europe: Moscow is usually snowbound for months in winter, but it doesn't actually get many heavy falls.