Is Penny an alcoholic?

TL;DR: Q: “Is Penny an alcoholic?” A: No.

Longer answer: she might have (had) a problem of alcohol abuse

Unless the writers are hiding something from the viewers, we see that Penny enjoys drinking in company, her favourite alcoholic beverage seems to be red wine, but she also drinks cocktails and the hard stuff to help her relax.

1. S4 E07, The Apology Insufficiency

Sheldon: I wronged Howard, and he won’t accept my apology.

Penny: Oh, right, that. Yes, Leonard told me. Sorry, honey.

Sheldon: Penny, you face failure on a daily basis. How do you cope?

Penny: I drink.

2. S5 E01, The Skank Reflex Analysis

Scene: Penny’s apartment. Penny has a bottle of wine. She can’t find a glass. Finds a measuring jug and uses that instead. There is a knock on the door.

Penny: Coming. Yup, that’s good. Wine glasses should have handles.

Amy: Keeping accurate track of your alcohol intake. Smart idea considering how trampy you get when you’ve had a few.

3. S6 E07, The Habitation Configuration

Sheldon: Penny, please, I’m on the horns of a relationship dilemma. And for the record, I had a full pubis of hair by the time I was 19.

Penny: And for the record, bleugh. So what are you drinking?

Sheldon: Well, it’s been a rough day. I usually go chamomile tea, but I don’t think that’s going to cut it.

Penny: You could have a Long Island Iced Tea.

Sheldon: Will that calm my nerves?

Penny: It’s calmed the pants off me a couple of times.

4. And in S7 E10, The Discovery Dissipation

Penny (listening in her apartment): You know, if we did a shot every time they said something embarrassing, this would be one hell of a drinking game.

Amy: A little early for alcohol, isn’t it?

Sheldon: You know, I don’t just say smart things about science. I also yodel. (Does)

Amy: I’ll get the vodka.

There was a previous beer drinking game with Amy Fowler in S5, E19 (The Weekend Vortex) but since she has been married to Leonard (S9 E01) it's exceedingly rare to see her alone with a bottle of red, or participate in these old drinking games. Although Sheldon reminds viewers more than once that Penny hasn't given up alcohol, and claims that Penny has a better relationship with white wine than with Leonard.

5. S8 E03, The First Pitch Insufficiency

Sheldon: I wasn’t making a joke, I was merely stating fact. Amy and I have a superior relationship to yours.

Leonard: You don’t honestly think that, do you?

Sheldon: Leonard, I assumed you knew. The ranking of relationships in our circle by quality is me and Amy, Howard and Bernadette, Raj and his girlfriend, Penny and Chardonnay, Penny and you.

Leonard: There isn’t any ranking. And if there were, we wouldn’t be at the bottom of it.

Penny: Yeah, and actually, I drink Sauvignon Blanc.

Leonard: That’s the part you have a problem with?

Penny: Relax.

Amy: There they go, fighting again. You’d never hear her talk that way to Sauvignon Blanc.

6. S8 E17, The Colonization Application

Sheldon: Do you want to say it?

Amy: Let’s say it together.

Together: We’re getting a turtle.

Penny: This is why I’ve been saying we should keep champagne on ice.

Sheldon: Sarcasm?

Amy: Yes.

Sheldon: Okay. That was tricky, ’cause when it comes to alcohol, she generally means business.

Leonard: Well, we’re, we’re very happy for you.

The website Web MD.com lists the following symptoms of alcoholism.

Signs of alcohol abuse

1. You have problems at work or school because of your drinking, such as being late or not going at all.
2. You drink in risky situations, such as before or while driving a car.
3. After drinking, you can't remember what happened while you were drinking (blackouts).
4. You have legal problems because of your drinking, such as being arrested for harming someone or driving while drunk (intoxicated).
5. You get hurt or you hurt someone else when you are drinking.
6. You keep drinking even though you have health problems that are caused or made worse by alcohol use, such as liver disease (cirrhosis).
7. Your friends or family members are worried about your drinking.

Signs of alcohol dependence or addiction

8. You cannot quit drinking or control how much you drink.
9. You need to drink more to get the same effect.
10. You have withdrawal symptoms when you stop drinking. These include feeling sick to your stomach, sweating, shakiness, and anxiety.
11. You spend a lot of time drinking and recovering.
12. You have given up other activities so you can drink.
13. You keep drinking even though it harms your relationships and causes health problems.

Other signs include:

14. You drink in the morning, are often drunk for long periods of time, or drink alone.
15. You change what you drink, such as switching from beer to wine because you think that doing this will help you drink less or keep you from getting drunk.
16. You feel guilty after drinking.
17. You make excuses for your drinking or do things to hide your drinking, such as buying alcohol at different stores.
18. You worry that you won't get enough alcohol for an evening or weekend.
19. You have physical signs of alcohol dependence, such as weight loss, a sore or upset stomach (gastritis), or redness of the nose and cheeks.

Penny has has never expressed the desire to give up drinking, she enjoys drinking, she enjoys the buzz that alcohol gives her, but there have been times in the past when she turned to alcohol for comfort and to numb the pain of a broken heart. In a 2011 comic-con interview, Kaley Cuoco said, “Penny has more baggage than any of the guys, let's be honest. That's the truth of it all.”

The drinking helps to dull the pain, to lighten the baggage temporarily; but is she addicted? No.

Perhaps the only significant scene where Penny's drinking lands her in deep trouble is in S4 E24, The Roommate Transmogrification

7. The Roommate Transmogrification (YouTube clip)

Penny and Raj are in Sheldon's apartment, drowning their sorrows when a sozzled Penny confesses to Raj

“I screwed up. Leonard’s a great guy. Never should have broken up with him.” The next morning sees Penny waking up naked beside Raj.

Penny (waking up): Oh, God. Oh, God! Oh. Okay, look. THIS. NEVER. HAPPENED. Do you understand me?

Raj: (nodding)

Penny: Really?! Still can't talk to me?!

[...]

Penny: [after Penny tries to sneak out, but finds Sheldon, Leonard, and Howard in the living room] Damn.

Leonard Hofstadter: What is going on?

Penny: [laughing nervously] Oh, it's, it's not what it looks like.

Sheldon Cooper: What does it look like?

According to Web Md.com guideline, Penny's blackout is the only sign that indicates she may have a drinking problem.

On the other hand, that scene was filmed way back in season 4 and one swallow does not make a summer. There are no other scenes from The Big Bang Theory that hint at any alcohol addiction. Penny enjoys a laugh, she enjoys going out with the girls in bars and restaurants, nowadays she is more like a social drinker. In fact, since she has married Leonard Hofstadter, Penny's character appears to be happier and more self-confident than ever before.

Within the show we have Leonard pointing out to Penny that she might have a problem in the episode The Locomotive Manipulation (Season 7, Episode 15).

The dialogue goes like this:

Bernadette Rostenkowski: You sure you guys don't want to come with us to Napa. You can probably still get a room.

Penny: No. I think we'll just have a quiet weekend at home.

Leonard Hofstadter: Of course I'm not sure it's a great idea to take Penny where wine comes from. What? It's a joke. Oh, come on. We bust on each other. I wear dorky glasses and you might have a problem. It's all for laughs.

Penny: That would really piss me off if I didn't have a buzz going on.

In Season 7, Episode 2, The Deception Verification, when Sheldon goes to Penny's apartment and and asks her why there are two glasses of wine on the table, we have this dialogue:

Sheldon Cooper: That’s curious. If there’s no one here, why are there two glasses of wine on the table?

Penny: Oh. Well, you know, I, I’ve got two hands and a bit of a drinking problem.

Sheldon Cooper: Of course. Ask a silly question.

Penny: Oh.

Penny is joking, but Sheldon calls her out on her problem, somehow... in a Sheldonian way