I've been through this recently, and the decision path was fairly straightforward:

Q1: Do the rights-holders of the media you will be streaming demand that you use DRM?

If yes, you must choose your DRM provider, by a mix of technical consideration, pricing, and negotiation with your content providers. This will then force you to use either Flash or Silverlight on the desktop, and a native application on mobile. HTML5 DRM is not available now, and may never happen.

Q2: If no to Q1, do you need to support low-latency streaming for specialised applications such as auctions or gaming?

If yes, you'll probably need a custom browser plugin.

Q3: If no to Q2, do you need to support older desktop browsers?

If yes, you will need to provide a Flash fallback.

Q4: Congratulations. You should use HTML5.

I've been through this recently, and the decision path was fairly straightforward:

Q1: Do the rights-holders of the media you will be streaming demand that you use DRM?

If yes, you must choose your DRM provider, by a mix of technical consideration, pricing, and negotiation with your content providers. This will then force you to use Flash or HTML5 DRM on the desktop, and a native application on iOS. You are unlikely to cover all target browsers with a single technology.

(This answer was originally submitted in 2013 when Silverlight was a feasible option for DRM. Silverlight plugin support has been removed from Chrome, and HTML5 DRM is now widely available.)

Q2: If no to Q1, do you need to support low-latency streaming for specialised applications such as auctions or gaming?

If yes, you'll probably need a custom browser plugin.

Q3: If no to Q2, do you need to support older desktop browsers?

If yes, you will need to provide a Flash fallback.

Q4: Congratulations. You should use HTML5.