All the tales of The Danish Fairy Book, edited by Clara Stroebe, are here. Some words are replaced by other words that are understood more easily, and the language is somewhat updated, British English here. Besides, the order of tales is shuffled so that the friendlier ones appear first. There is a note link at the end of each tale. If you find the name Grundtvig in the note of a tale, suspect the tale has been edited by him, for he used to do such things. A tale that has Grundtvig as its source - suspect an up to heavily editing hand in it; that is to say, that it is a retelling.
Grundtvig's fairy tales contain long embellishments, made-up remarks og repetitions, and strains from one or several variants to make the tale fuller and more nuanced. Moreover, out of the tales he got, he construed what he thought were original forms, and to this end he changed the settings back to medieval times with knights, tournaments, castles and more that were not in his sources. Some may say he went too far. [◦The Royal Danish Library: "Svend Grundtvig's Retellings" (In Danish)]
Be that as it may, a retelling can still have entertainment value. A translation of retellings may too. Professor Timothy R. Tangherlini has edited a contemporary English translation of Danish folktales (2014) to compare styles with.
Here is a good part of the romanticising preface by Frederick H. Martens, with practically no additions. Examples he gives, have been left out below.
These fairy-tales of the Danish North are the simple and direct narratives of the fishermen, peasants, seamen, huntsmen, shepherds, smiths and humble toilers of the land of Hans Andersen, faithfully presented in the traditional forms handed down from father to son through the centuries. They have the beauty and the charm of all that the fancy of a primitive people invents to express its longing for colour and radiance to illumine the dull greyness of everyday life. Out of a wealth of available material, stories of the most diverse nature have been chosen; stories in which the hero awakens the instant sympathy of the reader, either by noble self-sacrifice or by brave persistence, or by his sheer good luck.
Parts of what Martens writes above are unfit, since many of the "the simple and direct narratives" of Danish toilers were much edited by Svend Grundtvig (1824–1883). When he edited and published three volumes of Danish folk tales, he made up older, chivalrous settings and so on for several of the best known Danish tales. A purely editing hand had better refrain from adultering tales. Better point out that Grundtvig was a reteller among other retellers, like the Grimm brothers, Asbjørnsen and Moe and others. His notes about what sources he had used, were not included in the printed works, but they exist (Danish source link above).
Once on a time there lived a man and a woman who had an only daughter. One day a suitor came to call on her, so her mother sent her down into the cellar to draw some mead from the hogshead. When she had turned the spigot, it happened to occur to her, "Now if I get him and I'll see that I get him and we have a little boy and our little boy has a headache what will I wrap him up in?" And the jug stood under the spigot and the mead ran, and soon she was sitting in it up to her ankles.
Then her mother came down and asked, "Why are you sitting there?"
"Oh, I'm sitting here and thinking over that if I get him and I'll see that I get him and we have a little boy and our little boy has a headache what will we wrap him up in?"
"Yes, what will we wrap him up in?" said the mother. And the jug stood under the spigot, and soon they were sitting in the mead up to their knees. Then the man went down in order to see what had become of them.
"Why are you sitting down there?" he said. So his wife answered, "Oh, we are sitting here and thinking over that if she gets him and she'll see that she gets him and if they have a little boy and their little boy has a headache what can they wrap him up in?':
"Yes, what can they wrap him up in?" So he sat down beside them. And the jug stood under the spigot, and the mead ran and soon they were sitting in the mead up to their waists.
Now when all three failed to come back the suitor himself went down at last, and coming up to them asked, "Well, why are you sitting here?"
And then the man answered, "Oh, we are sitting here and thinking over that if she gets you and she'll see that she gets you and you have a little boy and your little boy has a headache what will you wrap him up in?"
But when the suitor heard that he went his way and never came back.
There was once a man who had three sons. When he lay on his deathbed, he called them to him, and told them he had nothing to leave them save an orchard, which they were to divide among themselves. One of the trees of the orchard bore the apples of health; but which tree it was, and where it stood, that he refused to tell them.
So the man died, and his sons went about dividing their heritage. But the youngest son was so small that he was not allowed to be present when the division was made. The two older brothers divided the orchard between them, and a single tree in the middle of the orchard was all they allotted their little brother. They gave him no land at all; for they thought that if ill luck would have it that his tree bore the apples of health, why, they could still gather up the apples which fell from it on the ground that was theirs.
One day it was noised about that the princess of that country was dangerously ill, and that the king had promised the princess and half the kingdom to whoever could restore her to health. So the brothers decided to try their luck without delay. The oldest at once ran into the orchard, picked an apple from every tree, laid them in a basket, and betook himself to the castle. The way led through a great forest, and as he entered it he met an old woman. "Good-day," said the old woman, "and what might you have in your basket?"
"Frogs and toads," said the young fellow, "and what business is it of yours?"
"Frogs and toads it is and shall be," said the old woman and continued on her way.
The young fellow went on and came to the sentry before the castle.
"What do you wish, my son?" the sentry asked.
"I have apples of health in my basket, and I want to go into the castle and cure the princess."
All well and good, but first the sentry would take a look at the basket. And when they lifted the cover it was swarming with frogs and toads. So the guards gave the young fellow a good beating and drove him off.
In the meantime the second brother had run into his half of the orchard, and had picked a basket of apples from all his trees. When he came to the forest he met the same old woman, who bade him good-day and asked him what he had in his basket.
"Snakes and worms," he answered, just as gruffly as his brother.
So she said to him, "Snakes and worms it is and shall be," just as she had to his brother. He came to the guard and asked to be admitted with his apples of health; but when they lifted the cover of his basket, the most disgusting snakes and worms came crawling out in heaps. And so they gave him a sound beating, just as they had his brother.
At last the youngest brother determined to try his luck. He picked some apples from his tree and set forth. In the forest he met the old woman.
''Good-day," said she, "and what might you have in your basket?"
"Good-day to you," said the little fellow, pleasantly, "I have apples of health in my basket."
"Apples of health it is and shall be," said the old woman, and passed on.
The little fellow went his way, and when he came out of the forest, the road ran along the seashore. There he saw the waves cast up an enormous pike on the beach, who lay there gasping for breath.
"You poor, unfortunate fish," said the little fellow, "I'll help you," and he took him and threw him far out into the water. Then the pike at once thrust his head out of the water, and cried, "I thank you. If you are ever in difficulties and I can help you, call me and I will come."
The young fellow went on and soon saw a raven and a swarm of bees who were fighting, and both sides having a hard time of it. So he went to them and told them how foolish they were to quarrel with each other, since each could fly in whichever direction they wished. That appeared sensible to them, and the raven and the swarm of bees, as they flew away from each other, both called after him, "We thank you for your good advice. And if you are ever in difficulties, and we can help you, call on us and we will come."
The young fellow went on and came to the sentry. "What do you want, my son?" asked the sentry. "I have apples of health in my basket, and would like to enter the castle and cure the princess."
Yes, that was all very fine, but first they would take a peep into the basket supposed to contain the apples, for they had already seen all sorts of curious things that day. But the basket was really full of beautiful apples, and when the young fellow gave the sentry a couple of apples to sample, he felt so bright and chipper that the boy was at once taken to the king, and by the king to the princess. He gave her a few apples, and when she had eaten the first she was able to raise her head from the pillow, and when she had eaten the second she was able to sit up in bed, and when she had eaten the third, she jumped up and danced about the room.
The king's heart was filled with joy, and he said that now she must marry the little fellow. But that not suit the princess. He would be a very trifling husband for her, was her opinion. So she told her father that any man who wished to marry her, ought to have made a name for himself; and if she was to take the little fellow, he ought to bring her, first of all, the ring that the king had dropped in the sea twenty-four years before.
The king told the young fellow what she had said, and the latter looked serious; but then he remembered the pike, ran down to the seashore, and told him of his trouble. The pike at once dived beneath the waves, and soon returned with the ring, with which the lad went back to the castle with a light heart.
The king received the ring with great surprise, went to his daughter and said, "You know what was promised the man who should cure you, so now let there be no more talk, but marry him."
The princess, however, said that would not do, she must have a husband who could provide her with a castle as large and splendid as the one in which she was living, and it must be built of wax, and shine like clear gold.
The king went back to the lad and told him what was demanded, and at first his face fell. But then he remembered the bees, and he ran out and told them his trouble. So they promised to do what they could for him. And on the following morning, when the people awoke, there stood a castle of wax, as large and splendid as the royal castle itself, and gleaming like clear gold.
Then the king went to the princess and said, "Now I can grant you no more time; now you must take him. You can see that he knows more than most people."
The princess had been much surprised at what she had seen; but still she was not satisfied. She insisted that the lad bring her the three oldest firebrands in hell. If he could only bring her those she would ask nothing further, but would gladly marry him.
The king was angry, yet he gave in to his daughter, and told the lad what she demanded. At first he felt very sad; but suddenly he remembered the raven, whom he had once helped, and he called him and told him his trouble. The raven promised to do what he could, and soon returned with the three brands. The lad took them, ran as fast as he could to the castle, and threw them into the princess's lap. They at once flamed up and she was well-nigh choked by the fire and smoke.
So she jumped up, very much frightened, and ran to the lad; for now she had no further objections to becoming his wife. And then they were married and he received half the kingdom as her dower.