Russian women are very independent and, in general, you won’t attract attention by travelling alone. That said, it’s not uncommon for a woman dining or drinking alone to be mistaken for a prostitute. Sexual harassment on the streets is rare, but a woman alone should certainly avoid ad hoc taxis at night – have one called for you from a reputable company.
Stereotyping of gender roles remains strong. Russian men will also typically rush to open doors for you, help you put on your coat and, on a date, act like a ‘traditional’ gentleman. (In return, they may be expecting you to act like a ‘traditional’ lady.)
Russian women tend to dress up and wear lots of make-up on nights out. If you wear casual gear, you may feel uncomfortable at a restaurant, a theatre or the ballet; in rural areas, wearing revealing clothing will probably attract unwanted attention.