I would suggest that the sonnet form is one of the popular and enduring forms of poetry -- just over 400 years of tradition there! The form of the poem helps to structure the meaning of the ideas. There are two traditional forms of sonnets -- English and Italian. Either form is 14 lines long and follows an identifiable rhyme scheme. The Italian structure is an octet (8 lines) and a sestet (6 lines). The poem doesn't have stanzas, but there is usually a single premise, argument or question posed in the first 8 lines that is then concluded in the final 6 lines. There is also a different rhyme pattern used in the 2 sections -- traditionally abbaabba cdecde -- but there are also many variations. English sonnets use a slightly different structure for the 14 lines. These are usually 3 quatrains (4 lines) and a final couplet (2 lines). In these poems, there are three thoughts and then a final concluding point made in the final two lines. The quatrains could be 3 examples, 3 observations, 3 arguments, etc. Frequently, each quatrain is a single sentence. There is usually a different rhyme set for each quatrain: abab cdcd efef gg -- but again, the rhyme patterns can vary.
The sonnet provides structure, and the reader of the sonnet can use the structure to help understand the poem. Poets use this form because it creates a built-in system to connect writer to reader. From there, poets can use any number of other literary techniques and figurative language to convey their themes.
If I understand your question correctly, I would suggest that what is popular and recurring in terms of poetry would be difficult to ascertain as different people are drawn by different kinds of poetry, just as they are with songs, novels, movies, etc. Art speaks differently to different people based upon their preferences and personal experiences.
In my opinion, I believe that the most popular kind of poem is narrative. This is a broad term that describes poetry that has a plot: in other words, it tells a story. These kinds of poems are traditionally some of the longest existing poems in our language.
Lilia Melani, at Brooklyn College, provides an in-depth and articulate explanation of the importance and characteristics of the ballad. The ballad (which is often associated with a song) is a very popular form of narrative poetry, specifically the folk ballad.
The ballads tell of love, death, the supernatural, or a combination of these.
Though generally short in nature and often anonymous, other kinds of ballads are fashioned after this early poetic form, including literary ballads.
Some of the most wonderfully romantic poems are narrative in nature, and would be considered literary ballads. For example, "The Highwayman," written by Alfred Noyes (1880-1958), tells the story of a highwayman and his love, Bess. This poem is about love, death and the supernatural. It is as haunting as it is beautiful.
And still on a winter's night, they say, when the wind is in the trees,
When the moon is a ghostly galleon tossed upon cloudy seas,
When the road is a gypsy's ribbon looping the purple moor,
The highwayman comes riding--
The highwayman comes riding, up to the old inn-door.
Over the cobbles he clatters and clangs in the dark inn-yard,
He taps with his whip on the shutters, but all is locked and barred,
He whistles a tune to the window, and who should be waiting there
But the landlord's black-eyed daughter--
Bess, the landlord's daughter--
Plaiting a dark red love-knot into her long black hair.
The beauty of these poems are not just in the content, but with the techniques they use to appeal to one's ear, for poems have a musical quality and are best when read aloud. (Before people could write, poems were passed down word-of-mouth, known as the oral tradition. People were enchanted in listening rather than reading.)
Examples of the devices most often used in poetry (often referred to as figurative language) are rhyme (internal and end), meter (rhythm), onomatopoeia, repetition, similes and metaphors. All of these assist the reader in drawing a picture in one's mind of the story being told (which is known as "imagery").
Form with ballads is often seen in four-line stanzas, that may be a pair of rhyming couplets, or a rhyme of the second and fourth line, etc., or may use stanzas longer than four lines (as with literary ballads). The rhythm and rhyme will provide the structural basis for the poem, and the literary devices employed (that appeal to the ear) will "musically" display the content of the poem to the reader/listener.