Use 301 redirects and drop the file extensions.
I can point you to some authoritative articles that will help you make your decision on how to approach redirects, URIs and SEO in general.
1) 301 redirects is considered the 'best-practice.' You can read the MOZ (widely considered and authority on SEO) article for more details.
2) As for best-practices on URLs (URIs), there is no reason to broadcast your file extensions - and what if you change, for example, from .php to another technology?
Ask yourself what value to the site visitors do file extensions in your URI's have? (Unless, of course they might have bad intentions - e.g. they like to exploit vulnerabilities in PHP, and that doesn't benefit your site at all!) - so drop the file extensions unless you can come up with a good reason to keep them.
3) Best-practices for the long-term
It is the the duty of a Webmaster to allocate URIs which you will be able to stand by in 2 years, in 20 years, in 200 years. This needs thought, and organization, and commitment.
The W3C's guidelines and recommendations will help you out a lot in setting up your site URIs using best-practices.
As for the impact on search engines, you rankings might stay the same, they might improve or they might drop.
This has much less to do with using 301 redirects than the fact that search engine algorithms change frequently.
For example, Google changes it's algorithms 500-600 times a year (about twice a day) - and the changes in the algorithms usually impact rankings, some times a little bit, and sometimes you will see a major change.
I always tell my clients that 'we follow industry best-practices and ethical SEO,' and IMO, this is always the way to go.
You can read the pros and cons of Black-Hat vs. White-Hat SEO for more details - although it's beyond the scope of your question, I think the frequent algorithm changes and the possible negative consequence of Black-Hat SEO make a strong argument to follow the recommended best-practices across the board - use 301 redirects and follow the W3C guidelines for URI's.
The following article, "Recovering from an Algorithmic Penalty Using 301 Redirects: A Personal Experience," reviews 2 case studies on 301 redirects and will help guide you through the process.
For your particular case, here is what I would recommend
Be consistent (i.e. if you use http://mywebsite.com/artist then stick to that format) - that's an example purpose only, I'll recommend a specific format (root-relative URL).
Even internal links that link to a file with the extension are problematic - for the same reasons. Link to '/directory-name' (/artist-name) with no trailing slash (it's not needed and you can configure your server to just treat this as a directory). I recommend using a root-relative URL for internal links (i.e. leave out the 'http://www.mywebsite.com' and use a forward slash such as '/artist-name') - this is based on the assumption that the directory or file is permanent in the file structure.
Keep your directory structure flat - "www.mywebsite.com/this-specific-artist" as opposed to "www.mywebsite.com/art/our-artists/this-specific-artist" - the shorter URL is, the easier it is for people to remember and type.
Take your time. Research and plan. There are a lot of great recommendations from trusted authorities in the links provided. Plan and think it through. Evaluate a) what are the shortest URLs; b) which URLs are most 'human readable'; c) what structure provides the greatest 'permanency'; and d) what directory and file naming conventions keep your options open to use a controller or any server-side programming language of your choice in the future (i.e. linking to file with a specific extension means you'll have to have either stick to that server-side language or do weird server configurations - so keep your options open and drop the file extensions).