I don't think I have favorite episodes per say, although if I had to name some favorites most of them would include River Song.
I do have favorite moments though and over time I've kind of strung them into a sequence of YT clips that I lookup in a particular order.
12th Doctor and Clara…
- Where the 11th Doctor calls Clara from Trenzalore, telling her to help the 12th Doctor. It's at this point that Clara truly sees the Doctor.
- Someone, 'Just save someone'. The ending scene of Fires of Pompei where Donna is telling 10 that he has to go back. Which dovetails into…
- The Girl Who Died, where the 12th Doctor remembers why he chose his face.
- The episode/scene where Clara tries to threaten 12 by throwing away the TARDIS keys. "Do you think I care for you so little that betraying me would make a difference?"
- Trap Street scene where 12 threatens to end Me and Clara argues that his reign of terror will end with the sound of the first crying child. Later, 12 explaining to Me to stay far from him because Clara was protecting her. "You'll find it's a very small universe when I'm angry with you!"
- Heaven Sent, the scene in the cloisters. "5 Billion years? Why would you do that to yourself?!"
River Song…
- End scene of Forest of the Dead, where she sacrifices herself.
- Voice Interface scene, "You will be dead in 32 minutes." Guilt, fish sticks and custard.
- Let's Kill Hitler, scene with Rory, Amy and River "Show me River Song". River saving 11.
- Ending scene of Let's Kill Hitler, River to Amy after being told she used up all her regenerations…"Mother, I had to try".
- Husbands of River Song scene with River and 12 on the cruise ship…"The Doctor is not stupid enough, or sentimental enough, and he is certainly not in love enough to find himself standing in it with me!" The whole scene for me is made with the look on River's face when she realizes the Doctor is right there with her.
- Darillium. "…how long does a night on Darillium last. 24 years."
- The episode and scene where Nardol tells 12 "Followed you from Darilium on the explict orders of your late wife River Song, warning, I have full permission to kick your ass."
I do have a couple of other scenes in between I like as well, but they don't really fit the sequence.
Rose…
I like Rose Tyler as well, but there's only one specific scene I gravitate to and it's when the 9th Doctor is gone and she's sitting in the cafe with Mickey and her mother. Her emotional and angry outburst shows just how much the Doctor has affected her and changed her.
Maybe all sappy stuff, but it's what I like about DW.
What I don't like…Davies making paper tigers and plot holes that mysteriously all seem to disappear. Why were we supposed to worry again if everything just magically resolves itself? Deus Ex Machina and even River Song fell for it.