Are there shibboleths specific to native Russian speakers?

A, an and the are all dropped. Using past tense with did (in my experience almost all non-native do this until they learn not to). Sometimes using she instead of he. Word order is not as important in Russian as in English. Missing prepositions

Russians I have met who have large vocabularies tend to stress words with more than two syllables in an idiosyncratic manner since they likely only ever read the words.
I have the same problem on rare occasions where I know a word, know how to use it but guess the pronunciation since I got it from literature.

More here

http://esl.fis.edu/grammar/langdiff/russian.htm

For example beginning learners often omit the auxiliary in questions or negatives: How you do that? / I no have it. The present simple is commonly used where the progressive form or perfect is needed: She has a bath now / How long are you in Germany?. In comparison with Russian the modal verb system in English is very complex. Mistakes such as Must you to work on Friday? / I will not can come, etc. are common among beginners. The lack of a copula in Russian leads to errors such as She good teacher.

Aside from the items pointed above, a well-educated native russian speaker often writes (and speaks) in incredibly long, almost Hemingway-ish, compound sentences, where you can barely remember what the beginning of the sentence was about. I'm not sure if it's primarily the influence of russian prose, or something about the language itself which causes the brain to produce the long sentences.

Russian and English languages have somewhat different structure of verb tenses. For native speakers of Russian it can often be difficult to correctly use perfect tense forms due to the influence of their mother tongue.
The grammatical concepts behind the correct usage of English perfect tenses can be very confusing to Russian speakers, so they tend to replace it with Simple Past tense for example (in case of Present Perfect or Past Perfect), or just fail to use it appropriately.

One thing that nobody seemed to mention is punctuation. It is of paramount importance in Russian, because it brings intonation across.

Here is a famous example from an old Soviet cartoon that is based on a tale by Hans Christian Andersen in which a little princess is asked to sign a decree of execution. Pay attention to the position of the comma.

• Казнить, нельзя помиловать!

  This means :

  Execute this person! Cannot pardon him!

• Казнить нельзя, помиловать!

  This means :

  Do not execute this person! Pardon him!

I guess you could argue that you can do the same in English like so:

Execute cannot, pardon! vs Execute, cannot pardon!

And this would make sense to an attentive English speaker, but punctuation tends to be not emphasized as much as spelling; as a result it will most likely be ignored or at the very least be ambiguous. I was just trying to illustrate the point that punctuation is so important that they made a cartoon for little children about it :-)

In fact it's so important that in Russia, Russian Language teachers usually give 2 grades for some written assignments: one for grammar and the other one for punctuation (it wasn't uncommon for me to get 5/3 ( or A/C in American equivalent) (I'm not a bad speller, but sometimes I can't get those punctuations signs right even if my life depended on it :-) )

To relate to this question though: you will notice that Russian speakers that finished at least 9 classes or high school in Russia will tend to use a lot more of , ; : " etc to bring extra nuances across, especially in run-on sentences because it's ingrained in the way language is taught. I see it with my Dad a lot. I've lived in the US for more than a decade now myself and I still tend to put commas in front of "that" in the middle of the sentence.

I am from Russia and I work at an international company so my colleagues and I have to use English all the time. There are really some common errors. The most difficult for us is to use articles properly. There are nothing similar to them in our native language. That is why we often use them where they are not needed and vice versa.

The second difficult part is using of prepositions. We tend to use those that we would use in our language if they were translated. For example, instead of at office we tend to say in office, instead of to London we often say in London. There are many other examples.

We don't have gerund in our language, so sometimes it is difficult for us to use it properly.

I can not agree with mplungjan that word order is not so important. It is important in any language and in Russian you can too change the meaning of a sentence if you change word order. Not always though, but in English it does not happen every time either.

There is also a rather big problem with sequence of tenses. In our language we do not have to do it. That is why we misuse perfect tense and even past tense forms often.

These are the most often encountered mistakes that I can spot when I talk to or read something from a native Russian speaker.