"The most common mistakes we see are candidates not proof-reading carefully enough and/or not researching the firm well before applying. Simple errors can occur from relying too heavily on spell-check or copy and paste functions. Lawyers have to be meticulous in their work as an error in a contract can change the meaning. We're looking for people who can demonstrate the same level of care. My advice would be to print out what you've drafted and read it on paper rather than on screen. Your eye is more likely to see the changes required. If necessary, ask someone else to check it for you.
The second main error is not researching firms well and making generic applications. We want to know why you'd like to work here and generic wording, such as "I would relish the chance to train at a leading firm who work on ground-breaking deals such as [insert deal from the news section of our website] isn't likely to answer to demonstrate this convincingly. Good research is likely to lead you to make fewer applications but of far better quality."