not much (Moscow, St. Petersburg or New York are much more dangerous). Only small problems, e.g. overpriced taxi, or a drunk man who knocked at our hotel room and said he wants money. But you can still get in serious conflicts with the authorithies. To avoid problems you must have all places you visit on your visa. Always wear the visa and passport in your pocket (my passport was checked by the police in the disco!!), and, for occasional ''fines'', have some dollars with you.
All people except citizens of the russian federation and citizens of a few other countries need visa to enter Russia. It's not a problem to get a russian tourist visa, but you need a visa support letter (from the hotel). Without this letter you cannot get the visa. Since 2002 it's mandatory to send the passport in original to the russian embassy, using of photocopies of the pass is not enough for visa processing, because the visa is sticked or stamped in the passport. Please check at your nearest russian embassy. If you are crossing the border, you have to declare all your money. Please don't forget this, you cannot take out money at departure which is not declared at entry. Please be alert at entry border crossing, that the immigration officer stamps the passport and the immigration card - if he does not, you will get problems. Your Hotel has to registrate the passport, so you have to leave it in the hotel. Make photocopies of the passport and the visa (!), so you can show by request a legitimation paper with your name in cyrillic letters, issued from a russian authority. That's very helpful. And - don't forget: Please don't extend your stay in Russia longer than the visa allows: You will get huge problems at departure and perhaps you have to pay a fine. And it's too late to leave the country at 1am, if the visa ends the day before....
It can be neccessary to use taxis. It's mandatory to arrange the price before starting the trip. If you call a taxi from a location out of Murmansk, arrange the price at telephone, especially for longer trips.
You will need russian money. You can change in the hotel or you can use the ATM-machine inside the hotel for getting money with your Credit Card. Please don't use ATM machines outside of your Hotel or of bank buildings - they may be unlegally changed. If you change at the private money changers at the street, be alert: Change only 50 USD maximum, take the russian money, count it in your own hands and after that handle out the Dollars. Don't change alone - the money changers are not alone, too. Don't take out a purse with money, let the money for the day in your pocket. And be aware of falsificated money.
A problem are the beggars in the streets. They will ask you for some change, many of them are drinkers. I cannot give you a recommendation, how you should do. It's the same problem like in many other big towns... Other people will ask you, if they can make a picture of you - with your own camera. Especially if you are making pictures with a camera on a tripod, many people will come. Never - never give your camera to other people - you have a big chance, that they run away with you photo supplies.