Possible explanations of what "the Spirit of Christ" is that it is simply God's Holy Spirit:

• that Christ had promised and
• that operates because Christ redeemed us and
• that is in alignment with Christ's will

There is therefore now no condemnation to those who are in Christ Jesus, who do not walk according to the flesh, but according to the Spirit. For the law of the Spirit of life in Christ Jesus has made me free from the law of sin and death.

For what the law could not do in that it was weak through the flesh, God did by sending His own Son in the likeness of sinful flesh, on account of sin: He condemned sin in the flesh, that the righteous requirement of the law might be fulfilled in us who do not walk according to the flesh but according to the Spirit. For those who live according to the flesh set their minds on the things of the flesh, but those who live according to the Spirit, the things of the Spirit. For to be carnally minded is death, but to be spiritually minded is life and peace. Because the carnal mind is enmity against God; for it is not subject to the law of God, nor indeed can be. So then, those who are in the flesh cannot please God.

But you are not in the flesh but in the Spirit, if indeed the Spirit of God dwells in you.

Now if anyone does not have the Spirit of Christ, he is not His. And if Christ is in you,

• the body is dead because of sin,
• but the Spirit is life because of righteousness.

But if the Spirit of Him (God's Spirit) who raised Jesus from the dead dwells in you, He who raised Christ from the dead will also give life to your mortal bodies through His Spirit (God's Spirit) who dwells in you.

Therefore, brethren, we are debtors — not to the flesh, to live according to the flesh. For if you live according to the flesh you will die; but if by the Spirit you put to death the deeds of the body, you will live.

For as many as are led by the Spirit of God, these are sons of God.

For you did not receive the spirit of bondage again to fear, but you received the Spirit of adoption by whom we cry out, “Abba, Father.” The Spirit Himself bears witness with our spirit that we are children of God, and if children, then heirs — heirs of God and joint heirs with Christ, if indeed we suffer with Him, that we may also be glorified together.

For I consider that the sufferings of this present time are not worthy to be compared with the glory which shall be revealed in us. For the earnest expectation of the creation eagerly waits for the revealing of the sons of God. For the creation was subjected to futility, not willingly, but because of Him who subjected it in hope; because the creation itself also will be delivered from the bondage of corruption into the glorious liberty of the children of God. For we know that the whole creation groans and labors with birth pangs together until now. Not only that, but we also who have the firstfruits of the Spirit, even we ourselves groan within ourselves, eagerly waiting for the adoption, the redemption of our body. For we were saved in this hope, but hope that is seen is not hope; for why does one still hope for what he sees? But if we hope for what we do not see, we eagerly wait for it with perseverance.

Likewise the Spirit also helps in our weaknesses. For we do not know what we should pray for as we ought, but the Spirit Himself makes intercession for us with groanings which cannot be uttered. Now He who searches the hearts knows what the mind of the spirit is, because He makes intercession for the saints according to the will of God.

-- Romans 8:1-27

To group it:

The Spirit of life

• No condemnation for those who walk according to it.
• Christ is not the Spirit of Life (Christ is God's Son, Christ became flesh, the Spirit is spirit)

The Spirit of God

• Believers are in the Spirit when the Spirit of God dwells with them.
• Raised Jesus from the dead
• Gives life to moral human bodies
• Those who are led by the Spirit of God are the sons of God

The Spirit of Christ

• The Spirit of Christ is required for salvation.
• Spirit of Christ is used interchangeably with the Spirit of life

The Spirit of adoption

• Bears witness with human spirit that believers are children and heirs of God

The Spirit

• The saints have the firstfruits of the Spirit.
• Helps the saints in their weaknesses.
• Makes intercession for the saints according to the will of God
• The Spirit knows our mind of human spirits

The human spirit

• Contrast the flesh
• Has a different mind than that of the flesh
• Bears the witness of adoption

Paul also wrote:

And because you are sons, God has sent the Spirit of His Son into our hearts, crying, “Abba! Father!” -- Galatians 4:6

However, according to Romans 8:14-15 this is the role of the "Spirit of God" and not the Spirit of God's Son. Unless Paul is contradicting himself, a possible explanation is that the "Spirit of God" is the same Spirit as the Spirit that is referred as "the Spirit of His Son".

For I know that through your prayers and the help of the Spirit of Jesus Christ this will turn out for my deliverance... -- Philippians 1:19

According to Peter, the Spirit of Christ already operated before Jesus had suffered.

Concerning this salvation, the prophets who prophesied about the grace that was to be yours searched and inquired carefully, inquiring what person or time the Spirit of Christ in them was indicating when he predicted the sufferings of Christ and the subsequent glories. -- 1 Peter 1:10-11 (ESV)

According to the author of Acts:

And when they had come up to Mysia, they attempted to go into Bithynia, but the Spirit of Jesus did not allow them. -- Acts 16:7

Note that the author said "the Spirit of Jesus" and not "Jesus' Spirit", just as "the Son of God" is not "God Himself", meaning "the Spirit" that empowers "Jesus" or "the Spirit" that was promised by "Jesus".

Word of Christ

Put on then, as God's chosen ones, holy and beloved, compassionate hearts, kindness, humility, meekness, and patience, bearing with one another and, if one has a complaint against another, forgiving each other; as the Lord has forgiven you, so you also must forgive. And above all these put on love, which binds everything together in perfect harmony. And let the peace of Christ rule in your hearts, to which indeed you were called in one body. And be thankful. Let the word of Christ dwell in you richly, teaching and admonishing one another in all wisdom, singing psalms and hymns and spiritual songs, with thankfulness in your hearts to God. -- Colossians 3:12-16

The "word of Christ" is not Christ personally himself. The author was writing in metaphors, for example "put on love", "bind everything together in perfect harmony", "peace rule" as well as "the word of Christ dwells...". This does not mean that Christ personally posses the believer like a demon. Instead, it means that believers have a responsibility to make Christ's word (teachings) part of their everyday lives.

In this context "the word of Christ" is the testimony or teachings of Christ that make Paul's disciples' faith possible.

But they have not all obeyed the gospel. For Isaiah says, “Lord, who has believed what he has heard from us?” So faith comes from hearing, and hearing through the word of Christ. -- Romans 10:17 (ESV)

Jesus' Spirit

However, it is worth noting that Jesus does have a human spirit like any other ordinary human otherwise he could not have been alive on Earth. But Jesus never promised his own human spirit to his disciples. The intention was that God's Holy Spirit should lead his disciples. There is a difference between "Jesus's spirit" and "the Spirit of Jesus". The latter means the Spirit that Jesus taught about. The same way as there is written you are the "sons of God". It does not mean we are gods but that God adopted us to be His sons.

Matthew wrote:

And Jesus cried out again with a loud voice and yielded up his spirit. -- Matthew 27:50 (ESV)

Mark wrote:

And immediately Jesus, perceiving in his spirit that they thus questioned within themselves -- Mark 2:8 (ESV)

The Pharisees came and began to argue with him, seeking from him (Jesus) a sign from heaven to test him. And he sighed deeply in his spirit and said, “Why does this generation seek a sign? Truly, I say to you, no sign will be given to this generation.” -- Mark 8:11-12 (ESV)

John wrote:

When Jesus saw her weeping, and the Jews who had come with her also weeping, he was deeply moved in his spirit and greatly troubled. -- John 11:33 (ESV)

After saying these things, Jesus was troubled in his spirit, and testified, “Truly, truly, I say to you, one of you will betray me.” -- John 13:21 (ESV)

When Jesus had received the sour wine, he said, “It is finished,” and he bowed his head and gave up his spirit. -- John 19:30