This article has been adapted from our sister site across the pond, Fool UK.

BP's (NYSE: BP) long, slow climb back toward respectability took another significant step on Thursday, with new chief executive Bob Dudley hinting that the company may resume paying dividends in 2011.

This will be good news for plenty of retired people (and pension managers) who rely on BP for its dividend income, as well as those, like me, who reinvest them for growth. Shares rose 2.5% following the news, and were up another 1.5% Friday morning.

You won't be surprised to discover that BP's shares are still well down on their pre-oil spill high. But they have steadily recovered from their post-spill low early July, as the leak at the Macondo well was finally plugged and then killed. At time of writing, they stand around $42. Is there more upside to follow?

BP still in deepwater
After the gush of bad news, there has been a slow trickle of good news coming out of BP. On Tuesday, investors queued up for BP's first U.S. bond issue since the Deepwater Horizon oil rig exploded in April. The five- and 10-year bonds attracted an impressive $12 billion worth of orders, four times the original allocation.

The company is also going through some much-needed restructuring. On Wednesday, Dudley announced a new safety and risk unit to ensure that all BP's procedures meet a common standard, and has divided its upstream unit into three sections, exploration, the development and production. It is also looking at whether its executive bonus scheme pays enough attention to safety.

This is only the beginning of its restructuring. The U.S. government forced it to sell $20 billion of assets to cover liabilities from the Macondo spill, and BP is now lining up another $30 billion worth of sales.

But Dudley still has a long way to go before he can convince people that this is a whole new safety-focused enterprise.

Personally, I like big companies that have seen their shares marked sharply down, but on balance should still survive. BP fulfils that brief, albeit with a major proviso.

BP remains a company on a knife edge. One more major accident, or even a minor one, could see it sliced and diced and sold off to its rivals. The problem doesn't even have to be BP's fault. You know what they say about giving a dog a bad name -- and few (if any) companies have as bad a name in the U.S. as BP.

This is particularly dangerous, given that BP remains heavily dependent on third-party contractors, and will carry the can for any errors they make. Any future attempts to blame third parties won't wash. Just look at how vigorously Halliburton (NYSE: HAL) is battling any suggestion of culpability in the recent crisis.

BP is now reviewing how it can exercise greater control over these contractors, but there's only so much you can do. Bear that in mind before buying BP on weakness.

Three strikes and it's out
After the Texas City refinery explosion in 2005, which killed 15 people, and the Deepwater Horizon spill, which killed 11 platform workers, BP cannot survive another fatal disaster -- not as an independent company. This is not a steady, low-risk stock. One error, and it could all end in tears -- or rather, in ownership by Exxon, Chevron, or possibly even a Chinese national corporation.

For that reason alone, it could be a long, long time before the BP share price returns to full health. Plus, of course, we still don't know the full cost of the cleanup.

Reputational damage is also harming BP's ability to make successful bids for future projects. It recently scrapped plans to bid for a license to drill in Arctic waters, fearing this would generate too much controversy. And it also admitted that it could take some time before it can resume deepwater drilling in the Gulf of Mexico. None of this will help profits.

Someone likes Hayward
The U.S. is the last country you want to subject to a major oil spill. BP will have lawyers crawling over it for years, with politicians providing covering fire. Dudley will be desperately hoping the company isn't accused of gross negligence, which will trigger punishing fines and another wave of court action.

It will be interesting to see whether restarting the dividend will provoke more anger in the U.S., and even pressure to reverse the decision.

BP's name may be mud in the U.S., but it is active in Russia. They may not like former chief executive Tony Hayward in Washington, but he has friends in the Kremlin.

You can be surer of Shell
Another factor driving the BP share price is the rising price of oil, with crude now up to $80 a barrel. If you want a play on oil, Royal Dutch Shell (NYSE: RDS-A) may be a safer bet. Its share price has risen steadily, and it is pumping out a more than 5.5% yield. BP will also give you exposure to oil prices, but a lot more uncertainty besides.

The oil giant's shares are on a positive trajectory right now, and that may continue if a healthy dividend payout is confirmed for the first quarter of 2011. Now could prove a good time to buy, but only if you're brave. 

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Fool UK contributor Harvey Jones owns shares in BP. Brian Richards prepared this article for publication on; Brian owns no companies mentioned.

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