I believe there is already a separate topic for this (I mean the typical grammar mistakes in native Russian speech).
I just want to sum up what grammar mistakes are extremely common.
- (100) грамм instead of (100) граммов - being discussed here;
- одевать/одеть instead of надевать/надеть (to put something on) - extremely common, although everyone knows it is wrong. "Одевать/одеть" means "to dress someone" (одеть человека), while "надевать/надеть" means "to put something on" (надеть куртку). However, so many people tend to say "одеть куртку"
- using "ложить" as an unprefixed imperfective verb. All the dictionaries and all the teachers say there is no such a verb, only "класть" (imperfective) and "положить" (perfective). Correct versions are: Она кладёт вилки на стол (She puts forks on the table) and Она положит вилки на стол (She will put forks on the table). But there are too many people who feel free to use "Она ложит вилки на стол" instead of "кладёт". As to me, it sounds completely uneducated.
- incorrect pronunciation of the stressed (!) vowel in forms of "платить/заплатить". Should be: он платит (he pays), он заплатит (he will pay), with stressed А. It's not uncommon to hear "он плотит, он заплотит" with a clearly audible stressed О. This one is a clear sign of low education, I'd say. I have almost never heard it from well-educated people.
- using "кофе" as a neutral noun. Previously we were taught "кофе" was masculine (despite its final -е), and it was considered absolutely wrong to use it as neutral. Now some dictionaries already admit that the neutral gender is an acceptable option for this word.
- pronouncing some forms of "звонить/позвонить" in the sense of "to call by phone" with the stress on the root vowel "о". Correct version: the stress falls on the endings in all its forms: звоню, звонишь, звонит, звоним, звоните, звонят, звонил, звони (and same with "позвонить"). Common trend: звоню, звонишь, звонит, звоним, звоните, звонят, звонил, звони (and same with "позвонить"). At least to me it sounds awkward. However, there are already some dictionaries which mark this new stress as an acceptable option.
They are the most frequent mistakes and/or disputable words.