Most of the social benefits went down the shitter together with the SU. Some of them still exist, but mostly on paper.
Russia has very limited welfare. Unemployment checks are hardly enough for buying food, but well, unemployment is relatively low, and most of the people who don't work are either picky or simply don't want to work because someone is paying for them. Healthcare is free, but first of all there is obligatory medicine insurance which takes away about ~4% of the salary, the second is that a lot of free services are either underfunded or of very bad quality, the third - pensioners get no free medicine on prescription like they used to, there is a money compensation for them, but it's often too small. Pension system is in permanent "reform" and deficit, while obligatory pension take aways is about 22% of one's income. An employer is supposed to pay them, but guess where he takes the money for it. Universities are free (you have good state exam results) or paid (you don't, but your parents have money). Schools technically are free and public, but they tend to milk parents for everything (from equipment to repair works).
The UR has been gradually removing social benefits from people, replacing them with money compensations, often inadequate, while introducing additional taxes that they mask as obligatory 'payments', 'insurances' and so on. The only welfare thing they introduced that I can recall was the one-time payment to the mothers of the second or more child (it's quite big, but there is a very limited list of things you can spend it on, like mortgage, the child education, etc).