We are currently looking at making a new website and wondered who everyone is currently using.

Integration with Linnworks is key for us and also the budget.

We do not have a large budget for a flash website or paying for a Magento designer. I have been looking at Shopify and Big Commerce to start with.

Any suggestions or feedback would be appreciated

We use Big Commerce - the integration with Linnworks is OK although we often use BC directly to modify listings.  Integration for orders is seamless and simple.  BC has improved dramatically over the last couple of years and unless there is something particular you need that isn't covered then I would recommend it to anyone.