I'm away from my books traveling, but this is my understanding.
All WWI Russian regular cavalry and steppe Cossacks used lances. That they were designated "hussars", "ulans", "dragoons" etc didn't make any difference at all other than tiny uniform distinctions. Battle formation was two ranks, the first lance armed and the second sabre armed for the lot.
The only cavalry in the Russian WWI Army that didn't use lances were those that wore Caucasian dress (Kuban Cossacks, Terek Cossacks and various mountain tribes).
In the Armed Forces of South Russia this was continued. If I recall correctly Mamontov notes being able to distinguish a Red unit from a White unit solely on that basis. I dare say some units struggled to find lances (they didn't like the British bamboo ones, but they used them) and some recruited men who took a while to be trained, but the standard was half lance and half sabre.
Out East, I can't say with as much certainty. I would expect any unit that was a reformation of a Russian Imperial unit, would want to be lance armed. Plus all Cossacks, as Cuprum says. (I've definitely seen pictures of Cossacks with them.)
The Reds used lances far less often, though they did clearly have some. I suspect that Cossack units that turned as a unit might continue to use them for a while, but it's probably easier to do without because it adds a point of difference. (Mine are half sabre, half carbine.) I would guess Bashkirs and other units formed from groups trained as cavalry before 1917 would use them. Most Red cavalry was commanded by men who had never seen service in the Imperial Army cavalry, so they would not be comfortable with them. Also they tended to frontally charge a lot less and use firepower more.