Basil and coriander grow well indoors. Lavender and rosemary can make quite big plants so do better in individual containers. Mint is a thug so also needs its own.
Parsley you can grow from seed but it can take ages to germinate - again I'd put it in its own pot. Thyme is happy in fairly dry gritty soil, as is marjoram. These are the only ones I've grown but other people will tell you about others.
I haven't had much joy trying to grow on those ready grown ones in supermarkets as they are forced and not strong plants.

Sure you can. All you need is a sunny window. Grow whatever you like. I have Italian Parsley, Oregano, Thyme. If you don't get enough sun for growing herbs, you can set up an indoor herb garden with 'grow lights'. Here's a video on just how to do it...

How I Tend the DIY Indoor Grow System
https://www.youtube.com/watch?v=XcVNk7wdE8Q

You can grow herbs indoors in pots on a preferably sunny windowsill - but don't expect to keep them going long term. Our climate here is lower in uv than that where most herb plants grow naturally, and that matters in winter particuarly. If you want to grow indoors because you have no planting space outside,then if you have the room,creating grow lights for the pots might work better, but with all the caveats listed in the first answer you got about individual plants. Parsley takes so long from seed (or often can) growing indoors on a windowsill can be tiresome... takes up space and doesn't produce much for ages.

Outdoors in pots or troughs, you can certainly grow rosemary, thyme, chives, oregano/marjoram and sage year round, they're hardy along with parlsey, though not necessarily evergreen. Also Fennel, but that would need a very deep pot for its tap root - Rosemary should be grown alone because its a shrub that gets quite large over time.

Parsley Many people find it hard to germinate parsley seed successfully. My own tried and tested method is to pour boiling water on the soil outside or on the compost indoors and then sow the seeds in to that immediately.
I have also heard of gardeners putting their seeds in hot water and leaving them to soak overnight with good results. Germination of the seeds takes approximately 3 weeks and planting to harvesting approx. 12 weeks.
If all else fails buy a small pot of parsley from the supermarket and transplant the seedlings in to the garden after the end of May. Plant one seedling in a six inch pot, preferably a deep pot like a recycled clematis pot. They like to put down long roots. It can be kept outdoors all summer and overwintered on the kitchen windowsill. Cut the biggest stalks of parsley first. Use scissors and take the whole stalk or the plants vigour will be lost. Parsley roots can also be used in cooking. Flat leaved parsley is mostly used in cooking and bread making.Curly leaved parsley is more often eaten raw or as a garnish. It will grow in partial shade or full sun but likes alkaline, free draining but moist soil. Mulch if it tends to dry out. Keep weed free and feed once a month with tomato food. Preserve leaves by freezing in a plastic bag. The frozen leaves are easily crushed for use. You can also dry them and store in an airtight jar.
My herb garden in a raised bed 7' x 3'. It has sage, 2 kinds of thyme, lovage (tastes like celery), Bronze fennel, sorrel (good for a sauce with fish or chicken) and burnet. In addition I put parsley and basil in there in summer. I have a bay tree and a rosemary shrub in the shrubbery. Lavender is grown in the sunken garden area. I grow Artemisia abrotanum (Appleringie or Wormwood) in the perennial border and use sprays of it in the kitchen to discourage flies and wasps. All of the above can be grown in pots.

No wondedr I've never had any success with parsley!

For my parsley, I follow Scotgran's method. In the Fall, I transplant it from the garden to a deep container and bring it indoors. It will continue producing straight through the winter. Don't discard the long taproots - it's nice in homemade chicken soup, especially on bitterly cold winter days like this one. Replace them with new plants in the Spring.