On the contrary to cygnus' answer, Scripture actually rejects the lie that an unregenerate person can be under the conviction of the Holy Spirit (Jn 14:17), since the Holy Spirit only leads people to Jesus Christ and His righteousness as the only ground of salvation (Jn 15:26, 16:13). To answer your question, the Holy Spirit does NOT AT ALL work in the reprobate. Certainly God is sovereign and has full control over all of His creation, including the reprobate, and God most certainly directs every single step the reprobate takes; however, the Holy Spirit is not involved.
No man can resist the motions of the Holy Spirit, and God does NOT show any love towards the reprobate. The blessings that the reprobate receive in this life only lead to their destruction (every single one of them). They are made to be destroyed, and everything God gives them only hardens them further. As it is written,
"When the wicked spring as the grass, and when all the workers of iniquity do flourish; it is that they shall be destroyed for ever" - Ps 92:7.
Vessels of wrath are fitted for destruction (Ro 9:22). Take for example what has been written of Judas, that wicked son of perdition,
"The Son of man goeth as it is written of him: but woe unto that man by whom the Son of man is betrayed! it had been good for that man if he had not been born." - Mat 26:24.
Did the blessings that Judas received in his life benefit him? Was it a blessing for him to be taught at the feet of the Lord Jesus Christ?
Keep in mind that Christians are to love their enemies, but loving their enemy's means that they don't speak peace to them. I don't love my enemies by telling them they are regenerate and fit for the kingdom, when in fact they prove themselves to be wicked blasphemers. I would hate them if I told them that they should be at peace with God when they are currently lost. Christians are commanded to love (i.e. do good unto) all men and seek to live peaceably with all men (i.e. Christians never seek to harm anyone); however, is this kindness a blessing to the reprobate? It is written,
"17 Recompense to no man evil for evil. Provide things honest in the sight of all men. 18 If it be possible, as much as lieth in you, live peaceably with all men. 19 Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord. 20 Therefore if thine enemy hunger, feed him; if he thirst, give him drink: for in so doing thou shalt heap coals of fire on his head. 21 Be not overcome of evil, but overcome evil with good." - Ro 12:17-21.
These loving acts were certainly not blessings to the reprobate, for they only increased their guilt towards God. God's people hate what God hates; therefore, they hate the false gospel of the unregenerate, as well as those who profess it; however, this in no way contradicts the command that Christians love their enemies. Christians do good unto their enemies, which includes telling them that they are unregenerate as well as praying for their enemies that God might save them (i.e. regenerate them). As it is written, "21 Do not I hate them, O LORD, that hate thee? and am not I grieved with those that rise up against thee? 22 I hate them with perfect hatred: I count them mine enemies." - Ps 139:21-22, and this "hatred" does no evil thing unto them.
God does not have a special salvific love towards His elect, and a different non-salvific love towards the reprobate. FYI, that is the definition of "common grace", which is a heresy. God is infinitely holy, righteous, and just; thus, He does not show grace or "love" at the expense of His holiness, righteousness, or justice. God only shows loves towards His elect through the work of Jesus Christ, and since the work of Jesus Christ is not universal in any sense; therefore, God does not show any love towards the non-elect reprobate. The so called blessings that fall upon the heads of the reprobate are only a curse onto them that cause them to flourish in order that they may be destroyed (Ps 92:7).
-jonas