I get a lot of women who come in and say they want to concentrate on their arms to get rid of their bingo wings so the first thing to note is that spot reduction is a myth. Women generally accumulate weight in certain areas such as the outer thighs, bum, stomach and arms whereas men tend to accumulate weight in the mid section, lower back and face (they can get double chins). Therefore, there is no point focusing exercise on the bingo wings only.
The primary thing to consider is diet rather than training. The accumulation of fat is generally caused by eating too many high glycemic foods such as white carbohydrates, refined sugars and processed foods. You need to stay away from them and drink plenty of water.
Another myth is that women are often worried about bulking up. However, resistance weight training will make you more compact and toned. You can do any resistance training: weights, Olympic or TRX suspension training. Those with injuries can do Pilates – Reformer Pilates enables you to add resistance – or yoga.
People who have bingo wings tend to focus on triceps but you need to focus on biceps, too (they are an agonistic and antagonistic pair of muscles; when one relaxes the other contracts) so you need to train them together to increase strength. You also need to train the lats (muscle under the arm connected to your back) with chin-ups, pull-ups and rowing. For the triceps, exercises include tricep extensions, cable pull-downs, and dips.
One thing I wouldn’t recommend is long distance running because you start burning muscle after 40 minutes. I would recommend a mix of circuits and resistance training (Nike has a good seven-minute app).