There are several factors that determine the rate at which a snowflake will melt as it falls to the surface:
(1) Perhaps the most important factor is the environmental temperature. The greater the maximum temperature within the elevated warm layer or the lower troposphere, the more quickly ice will melt. The elevated warm layer is a region of above freezing temperatures in the troposphere that is aloft from subfreezing temperatures. The heat transfer increases as the temperature gradient between the snow and the air increases. The depth of the above freezing layer also determines how much the snow will melt. If the above freezing layer is greater than 50 millibars in depth, all the snow will melt into rain in most cases.
(2) The initial temperature of the ice is another factor. As snow falls through air that is above freezing, the snow will first warm by rising from subfreezing to the freezing point. During this temperature increase the ice will remain frozen. To balance the energy transfer, the warming of the snowflakes will cool the surrounding air by conductional energy transfer, just as pouring warm water into cold water will warm the cold water and cool the warm water. The colder the snow starts off, the longer it will take to warm the snow to the freezing point. If the elevated warm layer or above freezing lower troposphere is shallow and only slightly above freezing, the snowflakes may be able to fall through it without melting whatsoever.
(3) At the same above freezing temperature, snowflakes that fall through dry air will melt slower than snowflakes that fall through moist air. Evaporation, sublimation and melting are cooling processes. When snow falls into above freezing air that is saturated, the snowflakes can not evaporate or sublimate much water vapor to the air. Since snowflakes in moist above freezing air can not absorb latent heat through evaporation or sublimation, all the latent heat must be absorbed by melting the snow. Snow that falls into dry above freezing air can absorb latent heat through evaporation, sublimation and melting. Since evaporation absorbs about 7.5 times as much latent heat as melting and sublimation absorbs 8.5 times as much latent heat as melting, the snowflake in above freezing dry air does not have to exchange as much energy with the environment by melting as a snowflake in moist air. The snowflakes will be able to cool the surrounding air while losing mass to evaporation/sublimation and will thus cause the snowflakes to melt at a slower rate in dry air as compared to moist air.
(4) Snowflake size also determines the rate of melting. A large snowflake will take longer to melt than a small snowflake. In a wet snow situation you may see drizzle falling along with the snow. The drizzle is small snow crystals that have melted. Heavy snow has a greater chance of surviving the elevated warm layer or the low level PBL temperatures that are above freezing than a light snow. A heavy snow will also be able to absorb much more latent heat in a shorter amount of time, especially if it falls through drier air. Heavy snow has a greater fall velocity. This limits the amount of time the snow is exposed to above freezing temperatures as it is falling.
(5) A day snowfall will be influenced by solar radiation while a night snowfall will not. Solar radiation can warm a snowflake just as solar radiation warms the surface. The influence of solar radiation can produce a wet snow even when temperatures throughout the troposphere are below freezing. Falling snow does have a very high albedo, which counteracts a large absorption of solar radiation but some of it is absorbed. Even the ground surface temperature makes a difference. A warmer earth surface will emit a greater amount of longwave radiation than a cooler earth surface. Water is a good absorber of longwave radiation emitted from the earth. Longwave radiation emitted from the earth's surface can warm falling snowflakes.
The next time it snows and temperatures are near freezing, think of the influence of the 5 mentioned factors on the snowfall.