This is funny. All people in here say that Russian has no dialects, well maybe it is true for Russia but remember that Russian is common in some other countries not only in Russia. I mean here in my place almost all people speak Russian in their everyday life but when people come to us from Russia they admit local Russian strange or special or just funny (ah, by the way I live in Belarus, my native town is called Гродно (Grodno) in Russian).But local people really think they speak perfect standard language so they are quite surprised hearing something like 'your talk is funny'
Well, I was sure I spoke standard too. But after I was told that I didn't I began paying attention to the way we were talking.
Here is something I noticed:

1) There is real difference between the age groups. I) Kids&teens&people of 20-50 speak close to the standard (at least closer than other groups do); II) adults of 50-60/70 (especially people who once moved from small places around Grodno like villages or farms) often have stronger accent and also more unusual grammar&vocabulary; III) people of 60/70- .... usually speak mix of local Belarusian dialect + standard Russian, or local Polish dialect + standard Russian, or local Belarusian + local Polish dialects + standard Russian (this is for people who were born in Grodno, or arond Grodno area). Though there are also people of different ages who moved here from Eastern Belarus or from Russia, they usually speak the standard Russian.

2) The groups understand one another though speaking different forms. For example I (the first group) easily understand my grandmother (the second group) and her sisters (the third group) and vice versa. While non-locals have some difficulties understanding the 2nd and the 3rd groups and rarely the 1st.

as for the most common local deviations (typical for all groups), here they are :

3) Какого сегодня числа? - Первого (второго etc.) января instead of Какое...? - Первое января.

4) Кудой мы пойдём? - Тудой! / Сюдой! instead of в каком направлении / как...

5) Споймать instead of поймать

6) Замного, замало, забольшой, заслишком etc. instead of очень много, мало, большой; и просто "слишком"

7) Спроси, или он пойдёт с нами. (Не знаю, или у меня получится. etc.) instead of Спроси, пойдёт ли он с нами.

8.) ихний (дом etc.) also ихняя/ихнее + noun instead of их (дом)...

9) снить/приснить: я приснила сон/ я снила сон instead of мне приснился сон/ мне снился сон

10) мне холодно в руки/ в ноги instead of у меня замёрзли руки/ ноги
Смеяться с кого-либо instead of смеяться над кем-либо
Or something like “Я не могу с тебя!”

11) нету (куда, что, где etc.) + verb: нету где поставить (сумку), нету что/чего одеть instead of некуда, нечего, негде + verb: негде поставить (сумку), нечего надеть

12) as for the nouns, sometimes Nominative = Accusative in plural form of the names of some animals like коровы, кони, козы, рыбы etc. : покормить рыбы, доить козы, пасти коровы instead of покормить рыб, доить коз, пасти коров

13) навязать (собаку) instead of привязать (собаку)

14) pronunciation
most of people cannot pronounce soft д and т sounds, they say soft affricates дз and ц instead like дзядзя, цёця instead of дядя, тётя
older people may also say fricative [ɣ] and hard ч instead of stop [g] and soft ч
unstressed vowels are often much clearer though оканье doesn't exist; people just say clear A instead of schwa
as for verbs, sometimes in the second person singular of the present tense they say [c'c'] instead of [шс'] like смеё[c'c']я instead of смеё[шc']я
м is usually hard in the words like семь, семья, people say like сем, семъя; the same thing is with the words like Вьетнам, which is said like Въетнам

15) gender
The most popular difference is собака, which is masculine instead of feminine: смотри, какой собака! instead of смотри , какая собака! Though it isn’t typical for all people, just many say like this. I use both of these depending on what the real gender of the animal is. There are some different-gender words but not many. Some geographical names like Свитязь (lake), Сож (river) are masculine instead of feminine
16) stress
Preposition is often unstressed, people say за рУку, на нОгу (or even на ногУ) etc. and also насквОзь instead of зА руку, нА ногу and нАсквозь; different stress when changing cases : вадУ, спинУ, тортЫ, тортА instead of вОду, спИну, тОрты, тОрта; different stress in many words: рАкушка, крапивА, красивЕе (or красивше at all), вербА, крЕмень, средствА, свеклА, искрА, хозяевА, теснО, щАвель, ивОвый, etc.
17) vocabulary
Well, there are some special words, for example шуфлядка, палендвица, паветка, складанка ; no one says ободок here, but everybody say обруч; I found the word базар funny, I don’t use it though some people do. I always say рынок. Long time I didn’t know what is половник (I thought it was дуршлаг) as my family says черпак. Many people don’t say свекла but use the word бурак. Plastic bag is пакет, or мешок, or торба ; торба is also non-plastic bag. Але, алеш, алиш…. I don’t know how to explain these mmm… just expressive stuff. When I said to a non-local girl “Алиш ты!” she asked “Я что?”

Well, maybe it is not a dialect, I don't know really what it is called.