How do I see my dota 2 MMR?

Yes, you can edit your profile, so that it is always shown, when you look at your profile.

When you click on "Edit Profile", there are two dropdown list, right next to the edit button.

(nevermind the fact, that I'm playing with the german client)

When there are no numbers, that means that you have not played enough games in either ranked solo or group (in my case, I did not play enough ranked solo, for my mmr to show up)

Here's how:

1. Open the new (Reborn) game client.

2. Click your name, top right.

   

3. Click Edit Profile, top right.

   

4. Choose the two MMR options:

   

5. Hit save!

There's now to boxes where you can showcase things, amongst which your solo and party mmr.

Another way is to click "Play Dota" then select the "Ranked Match" tab. You don't have to actually play a game, but it should tell you your MMR there.