What biblical evidence is there that our spirit only began in this life? LDS believe that our spirit was in existence before we are born here on earth. What biblical evidences are there that that is not the case?


  Genesis 2:4 These are the generations of the heavens and of the earth when they were created, in the day that the Lord God made the earth and the heavens, 5 And every plant of the field before it was in the earth, and every herb of the field before it grew: for the Lord God had not caused it to rain upon the earth, and there was not a man to till the ground. (KJV)


This implies that plants and herbs were in exhistance before they were placed on the earth. There also seems to be a double creation with Adam.


  Genesis 1:27 So God created man in his own image, in the image of God created he him; male and female created he them.
  Genesis 2:7 And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.


This implies that Adam was created spiritually, like plants and herbs, then created from the dust.

There are also other scriptures that support that man exhisted before this life. For instace Job 38:7, Eccl. 12:7, Jer. 1:5, John 9:2, Romans 8:29.

What evidences are there in the bible that our spirits were not in existence until this life?
I don't know if there are any particular verses in the Bible that specify souls do not pre-exist.  As Dick said, you are trying to prove a negative, which is difficult.
However, I did find a verse in the Orthodox Deutero-canonical text (referred to by others as the Apocrypha) that says there is no reincarnation:

A man indeed killeth through his malice: and the spirit, when it is gone forth, returneth not; neither the soul received up cometh again.
Wisdom of Solomon 16:14

It would stand to reason that if reincarnation is not true because a soul being in the place it goes to after life on earth does not come back to earth, then it follows that a soul simply does not come from there to here - it is a one way trip from here to there.