Isn't it the truth that meekness is rare in the world!
'I have said that when a sister speaks to you in the language or tone of passion or contempt you should answer her with sweetness. But I now say, that when ever the soul is disturbed, it is better to be silent; for passion will then make harsh expressions appear just and reasonable. But when peace returns, you will see that your language was altogether unjustifiable. St. Bernard says that anger draws over the soul a dark veil, which renders her incapable of distinguishing what is right from what is wrong.'
St. Alphonsus Maria de Liguori, 'The True Spouse of Jesus Christ'
'When a sister who has offended you comes to ask pardon, be careful not to receive her with a stern countenance, nor to show discontent or want of respect by your words or looks. If you give signs of dissatisfaction, you will violate charity, you will confirm the sister in her dislike towards you, and will give great scandal to the whole Community. You must, then, receive her with affection; and if, through humility, she fall on her knees to ask pardon, you should likewise go on your knees, and say to her: O my sister, why should you ask pardon of me? You know how much I love and esteem you. I ask pardon for having displeased you by my ignorance and negligence, and by my want of attention to you. Have pity, then, on me, and forgive me.'
St. Alphonsus Maria de Liguori, 'The True Spouse of Jesus Christ'
'When we have to reply to anyone who has insulted us, we should be careful to do it always with gentleness. A soft answer extinguishes the fire of wrath.'
'If we are angry, it is better to keep silence.'
St. Alphonsus Maria de Liguori
'Those who insult us and treat us contumeliously are anxious for a spiteful and sarcastic reply: the silence we then affect disheartens them, and they cannot avoid showing their vexation; they do all they can to provoke us and to elicit a reply, but the best way to baffle them is to say nothing, refuse to argue with them, and to leave them to chew the cud of their hasty anger.'
St. Ambrose of Milan
'Perhaps you may tell me that they are only going to insult me and that I should leave them alone and not bother about them. No, my good brethren. I can't abandon them: they are my own dear brothers and sisters. Tell me, if you had a very dear brother who was so sick that he was delirious and in his fever insulted you and said every foul thing in the world to you, would you abandon him? I'm sure you would not. You would feel all the more sorry for him and do everything you could for his recovery. Well, that is the way I feel about sinners. The poor creatures are just delirious. That makes them all the more deserving of compassion. I cannot abandon them. I have to work for their salvation and pray to God for them, saying with Jesus Christ, "Father, forgive them, for they do not know what they are doing or saying."'
St. Anthony Mary Claret
'Therefore, my beloved brethren, we should not look at these temporal things, but fix our attention on those which are eternal. Though affliction may come, it will have an end, though insult and persecution, yet are they nothing to the hope which is set [before us]. For all present matters are trifling compared with those which are future; the sufferings of this present time not being worthy to be compared with the hope that is to come. For what can be compared with the kingdom? or what is there in comparison with life eternal?'
St. Athanasius
'Margaret Mary Alacoque said: "I wanted no one to remember me save to scorn me, humble me and insult me; for indeed, that is all I deserved."'
St. Bernadette Soubirous
'To attain union with God, all the adversities that He sends us are necessary; for His only aim is to consume all our evil inclinations from within and from without. Therefore, slights, injuries, insults, infirmities, poverty, abandonment by friends, and relatives, humiliations, temptations of the devil, and many other things opposed to our human nature - all are extremely needed by us, that we may fight until by means of victories we have extirpated all our evil inclinations, so that we may feel them no longer. Nay more, until all adversities no longer seem bitter to us, but rather sweet for God, we shall never arrive at divine union.'
St. Catherine of Genoa
'"You have need of patience, that, after you have done the Will of God, you might receive the promise," says St. Paul; and the Saviour said, "In your patience possess you your souls." The greatest happiness of any one is "to possess his soul;" and the more perfect our patience, the more fully we do so possess our souls. Call often to mind that our Saviour redeemed us by bearing and suffering, and in like manner we must seek our own salvation amid sufferings and afflictions; bearing insults, contradictions and troubles with all the gentleness we can possibly command. Do not limit your patience to this or that kind of trial, but extend it universally to whatever God may send, or allow to befall you. Some people will only bear patiently with trials which carry their own salve of dignity, - such as being wounded in battle, becoming a prisoner of war, being ill-used for the sake of their religion, being impoverished by some strife out of which they came triumphant. Now these persons do not love tribulation, but only the honour which attends it. A really patient servant of God is as ready to bear inglorious troubles as those which are honourable. A brave man can easily bear with contempt, slander and false accusation from an evil world; but to bear such injustice at the hands of good men, of friends and relations, is a great test of patience. I have a greater respect for the gentleness with which the great St. Charles Borromeo long endured the public reproaches which a celebrated preacher of a reformed Order used to pour out upon him, than for all the other attacks he bore with. For, just as the sting of a bee hurts far more than that of a fly, so the injuries or contradictions we endure from good people are much harder to bear than any others.'
St. Francis de Sales
'If we do not feel within us a perfect patience, we have more reason to pity the grossness of our nature for being neither mortified nor dead to the things of this world, as we should be, than to blame those who load us with insults and ignominy.'
'We should not fear much the insults of this life, which are confined to words; were they all united they could not hurt a hair of our head.'
'They who load us with insults and ignominies give us the means of acquiring treasures more precious than any that man can gain in this life.'
'As much happiness as I felt on learning that the world had insulted you, I felt just as much pain at the single thought, that in your adversity you had sought aid and succor against the vexation and sorrow which it caused you.'
'If you wish to arrive speedily at the summit of perfection, animate yourself to a true love of shame, insults and calumny.'
St. Ignatius of Loyola
'The memory of insults is the residue of anger. It keeps sins alive, hates justice, ruins virtue, poisons the heart, rots the mind, defeats concentration, paralyzes prayer, puts love at a distance, and is a nail driven into the soul. If anyone has appeased his anger, he has already suppressed the memory of insults, while as long as the mother is alive the son persists. In order to appease the anger, love is necessary.'
St. John Climacus
'When harmed, insulted or persecuted by someone, do not think of the present but wait for the future, and you will find he has brought you much good, not only in this life but also in the life to come.'
St. Mark the Ascetic
'When you are insulted by someone or humiliated, guard against angry thoughts, lest they arouse a feeling of irritation, and so cut you off from love and place you in the realm of hatred.'
St. Maximos the Confessor
'During mental prayer, it is well, at times, to imagine that many insults and injuries are being heaped upon us, that misfortunes have befallen us, and then strive to train our heart to bear and forgive these things patiently, in imitation of our Saviour. This is the way to acquire a strong spirit.'
'They who have been exercised in the service of God for a long time, may in their prayers imagine all sorts of insults offered to them, such as blows, wounds, and the like, and so in order to imitate Christ by their charity, may accustom their hearts beforehand to forgive real injuries when they come.'
St. Philip Neri
'When your brother attacks you, whatever the insults are, if you get angry at him, you are getting angry without cause. Even if he were to pull out your right eye, and to cut off your right hand, if you get angry at him, you are getting angry without cause. Yet if he were to try to take you away from God, then be angry.'
St. Poemen
I couldn't find a quote I am long familiar with about the example of Jesus who, when insulted, did not reply, I am sure I will run across it sometime in future.

But I think the above give a good number of motivations for dealing well with insults. I especially think 'preparation beforehand' and prayer before and during is helpful. If you're not surprised by them, you're less likely to respond without thinking, in the wrong way. If you're praying through them, the sting might not sting deep enough to break your peace. And if you've change your perspective toward them being beneficial, you can surprise people by your atypical response if you respond at all.