Most scholars agree that the vineyards in this verse represent the relationship between the subjects, and the little foxes are those things which damage their relationship. God - who exists in perfect relationship with the other members of the Trinity - created us to also walk in perfect loving relationships with Him and others. And we would have no problem living this way except for the fall of mankind in the garden. Now we find the very thing for which we were created to be a constant battle.
In John 13:35, Jesus reminds us that "By this all people will know that you are My disciples: if you have love for one another." This is why we ought to concern ourselves with "catching" the little things that spoil our relationships. Walking in love towards God and others isn't just good advice, it is our calling and ministry. God wants us to live, empowered by the Spirit, as the people He created us to be in the garden of Eden. It isn't legalism - us earning our salvation - but rather us delightfully choosing to be what we should have been all along.
In our relationship with God, the little foxes can be "little" sins, things we dismiss as being "no big deal", but which God absolutely wants us to be rid of. A great example of this is "little white lies". The Bible teaches that the devil is the father of lies (John 8:44) and that "all liars" will be cast into hell (Rev 21:8), yet we casually toss falsehoods around out of fear, or to avoid difficult conversations. Or maybe it's gossip, which is like a delicious "dainty morsel" (Prov 18:8) to those who share it, but which Paul lists as a sin alongside wickedness, greed, and murder. (Rom 1:29)
The other little foxes that might spoil your relationships with both God and others are the "harmless" ways we waste our time, good things which rob of us the best things. For example, there's certainly no prohibition in the Bible against playing video games, reading, or watching TV, and those can be a fun way to pass some time, but when we choose to do that rather than to spending time in prayer, or loving our spouses, or nurturing our children, we rob our loved ones of something they deserve more than your right to have fun. Does your smartphone get more attention than your child?
Your relationships might also be suffering if you allow unimportant causes to alienate your acquaintances. The obvious example of this is politics. Do you push away people who need God's love and truth just because you vote differently? Or maybe it's denominationalism, where you would exclude someone from Christian love over the way your church performs certain rituals.
We were created in the image of a relational God to be relational people. Let's examine our relationships and catch any little foxes that might spoil them.