Are we supposed to be actively looking for a spouse, or waiting for God to bring a spouse to us?

At the risk of sounding noncommittal, the answer to both questions is
"yes." There is an important balance between the two. We are not to
frantically search for a spouse as if everything depends solely on our
effort. Neither should we be entirely passive, assuming that someday
God will cause a spouse to knock on the door, ring in hand, ready to
recite the vows.

When the time came for Isaac to take a wife, he took action (rather,
his parents did, according to cultural norms): they sent a servant to
actively seek a wife (Genesis 24). The servant had certain qualities
that he was seeking, and he bathed the process in prayer (verses
12-14). The Lord answered the prayer, and Isaac and Rebekah were
married (verse 67).

As Christians, once we know that it is time to start looking for a
spouse, we should begin the process with prayer. Committing ourselves
to God's will for our lives is the first step. "Delight yourself in
the LORD and he will give you the desires of your heart" (Psalm 37:4).
Delighting in the Lord means we find pleasure in knowing Him and
trusting that He will delight us in return. He will put His desires
into our hearts. In the context of seeking a spouse, that means
desiring for ourselves the type of spouse He desires for us and who He
knows will delight us further. Proverbs 3:6 tells us, "In all your
ways acknowledge him, and he will make your paths straight."
Acknowledging Him in the search for a spouse means submitting to His
sovereign will. Whatever He decides is best is what we want.

We should also be clear on the biblical characteristics of a godly
husband or wife, and we should seek someone who qualifies on a
spiritual level. It is important to understand these qualities first
and then seek someone who possesses those qualities. To "fall in love"
with someone and then discover he or she is not spiritually qualified
to be our mate is to invite heartache and put ourselves in a very
difficult position.

Once we know what the Bible says we should be looking for, we can
begin actively looking for a spouse, trusting that God will bring him
or her into our lives in His perfect timing. If we pray, God will lead
us to the person He has for us. It is important to wait on the Lord
and trust in Him (Proverbs 3:5).

Some people cite a verse in 1 Corinthians 7 that seems to say that we
should never look for a spouse. Verse 27 says, "Are you unmarried? Do
not look for a wife." This seems fairly straightforward, but the
context provides additional, important information. In verse 26 Paul
says, "Because of the present crisis, I think that it is good for you
to remain as you are." That is, in the time of persecution that the
early church experienced, Paul said it was better not to make big
plans, such as scheduling a wedding. Persecution hits a married couple
harder, and it is even worse if there are also children involved. Paul
repeats the sentiment in verse 29: "Those who marry will face many
troubles in this life, and I want to spare you this." In the same
context, Paul says, "If you do marry, you have not sinned" (verse 28).
So, seeking a marriage partner is not wrong.

Of course, it is not advisable to look for a spouse if God's will does
not include marriage. Sometimes God calls people not to marry at all
(1 Corinthians 7). There are people who "because of the kingdom of
heaven" have renounced marriage. Jesus says, "The one who can accept
this should accept it" (Matthew 19:12). In those situations, God makes
it clear that that is His will.