Ariston / Water Heater - When the city-wide heating is turned on in the winter (Dec. - Mar.) many homes/apartments have hot water since the heating system is water based (radiators). However, during the other months, if you want hot water, you need a hot water heater and not every place has one. It is one of the first quetions to ask about a place if you want hot water. (Sometimes a water heater is referred to as an Ariston, a brand name.) (Note: To have heat in the winter form the city system, the cost will probably be about US$30 for the entire 3 or 4 months. It may be included in the rent, but you should ask.)
Phone - Not every place has a working phone. If you want to connect to the internet from home you need to make sure the apartment/house has a working phone line. If your phone cost is not included in your rent, it should probably cost about US$2/mo. unless you're making long-distance or international calls.
Note on Cellphones: You can buy cell phones in Osh (better selection in Bishkek) and buy prepaid phone cards. Mobicards come in 200, 400, and 1000 amounts. 200 minutes cost about US$5 and last for 2 weeks. 400 minutes cost about twice as much and last for one month. After the time period expires, you need to buy another card even if you haven't used all your minutes. Sometimes your minutes carry over when you buy a new card and load the minutes in your phone. If you buy a phone in Bishkek and get a phone number there, it works fine in Osh. The cost, minutes used, etc., is usually the same throughout the country no matter where the call originates. The phones themselves are kind of expensive if you're used to getting them free back home when you get a phone plan. The cheapest phones are about US$60, but an average price might be more around US$90 or $100. It should come with a charger and new battery, but if the box included a hands-free set, it will probably be sold separetely even if it looks like it was originally included in the box. It's hard to tell if you're truly getting a new phone or an old one because they are usually displayed out of the box. If you get tricked into buying an old one you think is new, the phone usually works fine, it's just that the battery doesn't last as long. Most of the time, it's not a problem and you get a new phone. (Most phones from the US probably won't work in Kyrgyzstan unless they are tribands.)
Dual Gas/Electric Stove - This is a bit of a luxury, but it is extremely useful to have a stove that has both gas and electric burners. That way if the gas is turned off or the electricity is out you have another option. (Electricity goes out more often than gas so if you have to choose, a gas stove is best.)
Cable - Rumor has it that there is an official cable company in Osh now, but most places with cable get it from satellites that "borrow" signals. English channels usually include BBC, CNN, and maybe a couple more channels like BBC Prime. There are several channels in Russian, French, and Arabic, along with a mix of other languages like Spanish, Italian, German, etc. Usually, the cost is included in the rent and not charged separately since they really just need the hardware (satellite and box) and a "specialist" to decode the signals. If a box needs to be decoded again, it will probably cost about US$15 if your landlord doesn't cover it. If there really is an official cable provider, it would be worth looking into because they will probably have more English options (like the Star channel that shows American sitcoms), similar to what is available in Bishkek.
Refrigerator - Not all refrigerators have separate freezers so be sure to ask if it does (or check if you're looking at an apartment) if it is important for you to have one.
Electric Heaters - It is nice for an apartment to have an electric heater for the months before and after the city wide heating is turned on/off. If you need to buy one it will probably cost about US$45. You can buy cheaper ones, buy they might heat sections of a room v/s an entire room. If you see a portable heater in a place, make sure to ask if it will stay there. Sometimes the landlord may plan to use it himself/herself.
A Good Landlord - This is probably the most important thing to have. It is helpful to rent from someone who is used to dealing with foreigners and responsive to their needs. Otherwise, if you have a problem, it could continue being a problem for a long time or until you withhold rent, which could get ugly. You'd be surprised how small Osh can be as far as people knowing each other goes. Ask around about your potential landlord to see if anyone knows him/her to get their input. The number one reason foreigners change apartments/houses is usually due to "crazy" landlords. Some landlords believe they can continue to enter and use your place at will even though you're living there so you need to make it known that you expect them to call before they come over and it should infrequent. Other landlords may try to make you pay for things you should not. Make sure everything is spelled out in writing up front so you don't start paying for things you thought were included in rent (utilities, garbage collection, repairs, etc.). Note on Utilities: If you pay for utilities separately from your rent payment, gas (usually just the stove) will probably be about US$1/mo. and electric can be US$3 - US$12 (heavy usage - continuous water heater, electric heater, lights, tv, etc.). Water is about US$3 for a full year. It is helpful if your landlord agrees to pay the bills for you, showing you the bill or a receipt to justify the cost if you agree to reimburse him/her v/s including it in monthly rent. Otherwise, you may spend a lot of time waiting in line to pay your bills.
If you want something that's not there: If you want a washing machine (about US$300), a mircrowave (about US$100), an air conditioner (about US$350), etc., most landlords are willing to discuss buying one and perhaps splitting the cost. It usually results in you buying it up front and the cost is then deducted from rent.
Cost - Average apartment costs range from US$50 - $250. There are some that cost less, and some that cost more. A US$50 apartment will probably have a water heater, but few other amenities and will not be located in the city center. At $250, you'll probably have a washing machine, microwave, gas/electric stove, and A/C, and be located in the city center. For about US$150 you can probably have about the same, less one or two of the items above. Each apartment is different because apartments are individually owned. Renting houses generally costs more than apartments and will likely NOT be in the city center.