Is it possible to spam the bitcoin network with 'dust'? Someone could buy or mine bitcoins, and send them in small transactions all day to spam the blockchain.

Is this possible? And could this mean the end of the trust people have in bitcoin?

edit:

And what if someone makes bigger than 0.01 BTC transactions from one of his address to another all day. Making thousands of meaningless transactions, those totally spamming the blockchain. That wouldn't even cost much I believe. Is this possible?
Such transactions will be subject to fees.  You can read about the fee schedule here.  The fees apply if the transactions send a small amount of coins, or send the same coins over and over (giving the transactions a low priority score).  

The fee is set at BTC 0.0001 per 1000 bytes.  Thus someone who wants to spam the block chain with 10 MB will have to pay at least BTC 1 in fees.  That's about US$500 at today's prices, and it causes the block chain to increase in size by about 0.1%, which nobody will probably even notice.  If you're going to spend $500 on mischief, you'll get a lot more bang for the buck by spending it on eggs and toilet paper!