Can you provide reasons/evidence for why I should not trust Coinbase?
(I want to preface this by saying that I'm talking about people storing their savings in Bitcoin. If you don't care about what happens to your money, then it doesn't matter whether you trust Coinbase.)
The most convincing argument, I think, is that you're putting your faith in an entity that's not FDIC insured to hold onto your entire future and keep it safe. Your savings is your future. If you lose it, you lose your future. You'll have to exchange months or years of your life to get it back, as I now have to. There are many reasons that businesses fail, and it's important to never forget that Coinbase is a startup, and the fate of startups always hang in the balance.
There's no reason not to trust an entity that's ensured by the government (banks) with your savings. They won't vanish.
The only reason right now to trust Coinbase with a lot of money is if you're hoping the price will rise, as far as I can tell, because there's no other realistic application of Bitcoin right now unless you want to send some money to a friend, in which case you can just buy it as-needed. So it's essentially long-term gambling. Now, there's nothing wrong with that. I knew when I did it that there was a chance things could turn out horribly awry (though I don't think I fully appreciated how "down" the downside could be for me). But if you're going to do it, at least be smart about it: The coolest thing about Bitcoin is that you don't have to trust anybody but yourself to keep your coins safe. There are secure cold storage wallet solutions. Find one, use it. I've heard good things about Armory, for example.
You could counter that by saying "Oh, those are just drug addicts. Normal people aren't that dumb." To that, I can point at the example of the Mt. Gox collapse. There you have a case of people going from millionaire status to "looks like I'll have to find a job and keep it for a long time" status in the blink of an eye. One of the bigger losses was 4,700 BTC: http://www.reddit.com/r/Bitcoin/comments/1yv26o/gox_horror_s...
And lastly, you can't have it both ways. Either Coinbase is trustworthy enough to keep your personal savings safe, or it's insane to trust them with any amount of money that you care about not-vanishing.
You are asking the wrong question -- why should you trust Coinbase or anyone else?
The problem with Bitcoin right now is that it isn't convenient, even if you use these third party services. It's not convenient having to trust the services to provide you with convenience because if they fail, you fail.
But, ultimately, if you don't mind losing your money, it doesn't matter whether you trust Coinbase.
As far as I know (IANAL) there is no law or precedent which takes the value of stolen bitcoins into consideration while sentencing. There is also no history (AFAIK) of prosecution, of serious prosecution which takes the effort to follow up on stolen coins may have been "mixed" etc., in cases of BTC theft.
Given this, does not an online service (like Coinbase) which holds a large amount of bitcoin in "stealable" form present, to potential criminals, a lot of incentive to break in and take it all away? I, for one, wouldn't keep up to 5000 USD in such a place, at least till the time there has been some serious prosecution and proportionate sentencing of BTC thieves. But of course to each their own.
In April 2013, Mt Gox handled 70% of bitcoin trading [1] - they were a widely trusted, well known brand, operating in a stable legislative environment.
Coinbase today has what Mt Gox had a year ago.
Admittedly, coinbase also has some VC investors. If you believe, should they be hacked or robbed by an insider, that the VCs are going to throw good money after bad to compensate customers, you are much more optimistic than I am.