I have listed here the best arguments I have heard against the Soul Sleep doctrine:

1. Thief on the cross

First, there are several other places in Luke's gospel, where Jesus have used "Ἀμὴν λέγω ὑμῖν" (translated as "Truly I say to you"), but he didn't add today at the end of any of them! Those places are: 4:24, 9:27, 12:37, 12:44, 18:17, 18:29, 21:3, and 21:43

It is true that the oldest existing manuscripts of the New Testament do not contain punctuation marks, and the alternative punctuation is theoretically possible.

However, since they were both dying, there is no other time Jesus could make this statement than that very day. Furthermore, his statement is in response to a specific request, "Lord, remember me when You come into Your kingdom." Jesus told him He would enter God's kingdom immediately, in response to the man. 

Moreover, there is, in fact, one place in Luke (verse 19:9) where the word 'today' (SHMERON) comes at the beginning of the sentence of clause:

NIV Luke 19:9

Jesus said to him, “Today salvation has come to this house, because this man, too, is a son of Abraham."

2. Jesus preached To Spirits

Scripture says that Jesus preached to the spirits in prison. 

For Christ also suffered once for sins, the just for the unjust, that He might bring us to God, being put to death in the flesh but made alive by the Spirit, by whom also He went and preached to the spirits in prison (1 Peter 3:18,19).

There is no purpose in guarding unconscious spirits in prison or preaching to them. The fact that they were under restraint shows them to be conscious. 

3. Moses And Elijah

We also have the account of Moses and Elijah appearing at the transfiguration of Jesus (Matthew 17:1-8). The impression is that they are coming from a sphere of conscious life to converse with Jesus. There is no indication that they are awakening from some dreamless sleep, or from some sphere no better than the earth. 

4. Not Permitted To Talk

Paul clearly said that he was not permitted to relate his experience of the time he was in the presence of God. 

Paul was caught up into Paradise and heard things that are not to be told, that no mortal is permitted to repeat (2 Corinthians 12:4).

Then, we cannot know that Lazarus, for example, hasn't seen anything while being dead. All we know is that such an experience was not recorded in the scripture. Maybe, Lazarus was also not permitted to talk.

5. With the Lord, away from the body

The Bible records shows that in the New Testament, all of God's people who are in the state of death are away from the body and present with the Lord:

NIV 2Cor. 5:8

We are confident, I say, and willing rather to be absent from the body, and to be present with the Lord.

NIV Philippians 1:23-24

…I desire to depart and be with Christ, which is better by far; 24 but it is more necessary for you that I remain in the body

If Paul was asleep at death, how could he say that to die, and to be with the Lord, was far better than being in this body?

NIV 2 Corinthians 12:2-3                                                     

I know a man in Christ who fourteen years ago was caught up to the third heaven. Whether it was in the body or out of the body I do not know—God knows.3 And I know that this man—whether in the body or apart from the body I do not know, but God knows.

Final words:

The question whether to accept Soul Sleep or not is quite difficult, but my personal opinion is that the arguments supporting Soul Sleep have slightly bigger weight. If you are interested in the supporting arguments, you may take a look to this post.

The other answers include many of the quotes I would cite. Here is one I do not see listed:

When he opened the fifth seal, I saw under the altar the souls of those who had been slain for the word of God and for the witness they had borne. They cried out with a loud voice, “O Sovereign Lord, holy and true, how long before you will judge and avenge our blood on those who dwell on the earth?” Then they were each given a white robe and told to rest a little longer, until the number of their fellow servants and their brothers should be complete, who were to be killed as they themselves had been. (Revelation 6:9-11, ESV)

This does not flatly contradict soul sleep, but it does show that at least some Christians are awake for part of the time between their death and resurrection.

The Bible never described someone else's soul sleeping only to be awaken at the last judgment.

Rather, what the Bible records show is that in the New Testament all of God's people who are in the state of death are away from the body and present with the Lord. They are not sleeping but are at home with the Lord.

Yes, we are of good courage, and we would rather be away from the body and at home with the Lord

2 Corinthians 5:8 (ESV)

Paul himself expressed, not the desire to sleep, but to be with Christ.

21 For to me to live is Christ, and to die is gain. 22 If I am to live in the flesh, that means fruitful labor for me. Yet which I shall choose I cannot tell. 23 I am hard pressed between the two. My desire is to depart and be with Christ, for that is far better. 24 But to remain in the flesh is more necessary on your account.

Philippians 1:21-24 (ESV)

2 Corinthians 12:2-3 bolsters this interpretation:

2 I know a man in Christ who fourteen years ago was caught up to the third heaven—whether in the body or out of the body I do not know, God knows. 3 And I know that this man was caught up into paradise—whether in the body or out of the body I do not know, God knows—

2 Corinthians 12:2-3 (ESV)

God knows that a man can be with Him in the paradise. This is not sleep but a relationship! Take note that The thief on the cross didn’t sleep! Jesus promised him on that day that he will be with him in the paradise! (Luke 23:43).

What all this shows is that human death is not about "cessation of existence." Rather, it is about separation.

For as the body apart from the spirit is dead, so also faith apart works is dead.

James 2:26 (ESV)

James 2:26 says that death is a separation of body and spirit. Both do not undergo annihilation after death. Rather, both merely separated.

This one scripture powerfully attests the doctrine of "souls don't sleep":

who died for us so that whether we are awake or asleep we might live with him.

1 Thessalonians 5:10 (ESV)

Conclusion

We have cogently proven from the Bible that "soul sleep" teaching is unbiblical. The New Testament consistently shows us that souls do not sleep and that the metaphor 'sleep' used for those who are dead is merely a reference to their physical bodies appearing as if sleeping.

References:

http://www.gotquestions.org/absent-from-the-body.htm

https://carm.org/soul-sleep