There is a lot of misinformation around, so I’d like to qualify this by saying I am a Radiographer of 5 years experience, having worked with CT for 4.5 years, and am part way through a Masters in MRI.
CT scans for musculo-skeletal (MSK) indications are primarily for bony abnormalities. CT can often detect acute and/or larger muscular abnormalities, but it is somewhat dependent on the body fat content of the patient. CT images on patients with some fat layers between their muscles are easier to read. However while CT is the gold standard imaging modality for bony abnormalities, if there is no bony involvement, and no highly vascular structures such as a tumour or arterio-venous malformation (AVM), I would not recommend it.
The first option for muscular injuries, depending on location, should be Ultrasound (US). US is usually cheaper than MRI, it is faster, and in many cases it is just as sensitive. It is limited for deep muscles, as a lower frequency probe with lower resolution must be used. US is also limited when structures are hidden behind bone, because unlike CT and MRI, it cannot penetrate bone. For example in the shoulder, US is fantastic for imaging the muscles you can directly feel with your fingers, but can’t image the muscles that sit between the shoulder blade and the rib cage.
MRI is, when correctly performed, the gold standard for MSK imaging (bony abnormalities aside). It is generally the most expensive form of imaging, and the scans can take quite some time. But the general rule with MRI, is that it is good at imaging anything with a lot of water or fat, provided there is a difference in the water or fat contents between tissues. MRI is however highly dependent on the operator. There are dozens of different sequences an MRI can use, and the selection of the correct sequences is vital. Some centres will limit their protocol to simple series, so the scan is shorter and they can get the next patient in sooner. More scans equal more money. But if the correct sequences are chosen, MRI will generally provide more information on muscular injuries and abnormalities than any other form of investigation.
Summary
I would recommend you get an Ultrasound first. If it is enough to diagnose your situation, then you’ve saved a lot of time and money. If you aren’t able to have an MRI (many people aren’t, for safety reasons), Ultrasound is still a very good option for most muscular issues. However MRI is the gold standard for most muscular issues, if it is correctly performed. If the Ultrasound doesn’t yield enough information to inform your recovery, then an MRI is an excellent next step.