Herbs on the windowsill
What actually works indoors — The right herbs on a bright windowsill will outlast anything from a supermarket pot. The wrong ones will just slowly die.

The ones that actually work
Chives, mint, and parsley are the most reliable on a UK windowsill year-round. They tolerate lower light levels and cooler temperatures without sulking. Rosemary and thyme do well if you have a south-facing window with good direct light.
Basil needs warmth and sun — it's a summer herb. From October to March, most UK windowsills can't provide what it needs and it'll decline slowly. Coriander bolts fast at the best of times; sow a small amount every few weeks rather than trying to maintain one large pot.
Pots and drainage
Herbs hate sitting in water. Use pots with drainage holes — always — and empty the saucer twenty minutes after watering. A free-draining compost helps: multipurpose with a handful of perlite mixed in works well. Terracotta is good because it dries faster between waterings than plastic.
Most herbs are Mediterranean in origin and evolved in poor, dry, well-drained soils. Rich, consistently moist compost suits them far less than a neglected corner of a sunny terrace. Water less than you think. Let the compost almost dry out between waterings, then water well.
Harvesting properly
Cut from the top and outer stems, leaving at least two thirds of the plant intact. For basil, pinch out the growing tips to prevent the plant from flowering — once it bolts, the leaves become bitter and the plant starts to decline. Keep it bushy and vegetative.
With mint, cut it back hard every few weeks. It'll come back bushier and more vigorous than before. Mint in a pot will eventually exhaust the compost — repot it into fresh compost in spring, or divide and repot into two smaller pots.
Supermarket herb pots
The herbs sold in supermarket pots are grown fast and dense to look full for a week on a shelf. They're not meant to last. But with a little work, they can.
Split the rootball into three or four sections, pot each one into its own container with fresh compost, and leave them alone for two or three weeks to recover. Then treat them like normal plants. You'll get four times the herbs for the same price, and they'll outperform the original pot within a month.