Show, don't tell
I recently read an article––unfortunately I can't remember where––noting that "show, don't tell" is not so much a rule as a fashion that is now becoming passe. That could be due to the realization that some writers are taking this to the point of extreme tedium. True, if a character plays an important role in a story, for example, an overbearing mother, you will want to show us instances where the mother is being overbearing. But if the mother is dead or Mary wonders, just in passing, if her mother's overbearing personality might have influenced the way she raises her own children, you can simply say that. You don't need Mary to flashback 20 years remembering her mother's words on the night of the senior prom. That's just added junk your story doesn't need.
Avoid cliche like the plague
Okay, I threw that cliche in there just for a little humor, but some writers think avoiding cliche means never saying anything the same way others have said it. A character's eyes can't "well with tears." Instead, "pools of water form and hang suspended above his lower eyelids." What do all those extra words accomplish beyond telling us the the writer has oh so cleverly found a different way to describe a common phenomenon. We've all seen someone's eyes fill or well with tears. You really can't add any clarity to such a standard image, so just say what you need to say and move on.
Always use active rather than passive voice
This is one Hope mentions as well, and it is another rule that, taken too seriously, can ruin your writing. Again, this is partially a matter of fashion. Writers of bygone times often made more use of passive voice than we do today as do many writers outside the US. More important than sticking to the active voice is making sure the voice you choose fits the story and the time in which it is set, and varying your sentence structure. Just like anything else, now and again we need to slow the pace for our readers, and passive sentences are the best way to do that.
No run-on sentences
One of my favorite authors, Edith Wharton, wrote sentences that took up entire paragraphs. She wrote mostly about the Gilded Age in America, and I can't imagine writing about those times with anything other than her perfectly crafted sentences that swirled this way and that, circling back to where they started. Even Ernest Hemingway, king of the minimalists, whom we usually associate with the short, declarative sentence, in reality often used very long sentences peppered with conjunctions, especially in his descriptions. Here's an example from For Whom the Bell Tolls,
This officer shot two men as they lay and still they would not get up and he was cursing them and finally they got up, one two and three at a time and came running toward us and the train.
Speaking of rules, notice the unorthodox use (or lack thereof) of commas in that excerpt.
It's good to read a few magazines or books about writing to get a general flavor of the current writing "fashion." However, writers will learn even more by reading the actual work of contemporary writers as well as older writers whose styles could add a little variation to our own writing. By doing so, writers develop an ear for what works and don't need to rely so heavily on the "rules."